#6754432

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6661628);