#72287724

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#102
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#6358316	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#135 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=i,b^2=g,c^2=e,d^2=gij,e^2=i,f^2=h,g^2=ij,h^2=j,b^a=bg,c^a=cij,c^b=cgj,d^a=dgij,d^b=di,d^c=df,e^d=ej,f^a=fij,f^c=fh,f^d=fhi,g^a=gij,g^c=gij,h^c=hj,h^d=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287776
maximal normal subgroup of	#72287776

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