#72287771

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23029
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#490656	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8192 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,6,7)(3,11,14,8)(4,13,15,9)(5,10,12,16)(17,18)(19,23)(20,33)(21,26)(22,28)(24,32)(25,36)(27,35)(29,45)(30,39)(31,40)(34,41)(37,43)(38,47)(42,46)(44,48);(1,3,2,8,6,14,7,11)(4,5,9,10,15,12,13,16)(17,19,21,30,22,31,35,32)(18,23,26,39,28,40,27,24)(20,29,36,46,37,34,48,47)(25,42,43,41,44,38,33,45);(1,4,6,15)(2,9,7,13)(3,12,14,5)(8,16,11,10)(17,19,22,31)(18,24,28,39)(20,34,37,29)(21,30,35,32)(23,27,40,26)(25,41,44,45)(33,42,43,38)(36,47,48,46);(2,7)(3,13)(4,11)(5,10)(8,15)(9,14)(12,16)(17,20,35,48,22,37,21,36)(18,25,26,43,28,44,27,33)(19,29,30,46,31,34,32,47)(23,38,24,41,40,42,39,45)

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=i,d^2=gi,e^2=i,f^2=h,g^2=ij,h^2=j,b^a=bj,c^a=cg,c^b=cgij,d^a=dg,d^b=df,d^c=dfhij,e^d=ej,f^a=fij,f^b=fh,f^c=fhj,f^d=fhi,g^a=gij,g^b=gij,g^c=gij,h^b=hj,h^c=hj,h^d=hj

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287776, #72319504, #72319684, …
maximal normal subgroup of	#72287776, #72319504, #72319684, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.9);;
gap> SetPower(coll, 4, F.7*F.9);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);