#72287705

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	#23188
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#33748	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8293 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

784 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=efhj,d^2=egh,e^2=i,f^2=h,g^2=ij,h^2=j,b^a=bj,c^a=cf,c^b=cgij,d^a=df,d^b=di,d^c=dfgi,e^c=ej,e^d=ej,f^a=fh,f^c=fj,f^d=fh,g^c=gij,h^a=hj,h^d=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287720
maximal normal subgroup of	#72287720

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.5);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.5*F.6*F.8*F.10);;
gap> SetPower(coll, 4, F.5*F.7*F.8);;
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.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);