#6800808

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1397 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)(20,32,33,25)(22,24,35,27)(30,37,42,41)(31,36,43,38)(34,40,45,39)(44,47,48,46);(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,19)(18,23)(20,30)(21,29)(22,31)(24,38)(25,37)(26,28)(27,36)(32,41)(33,42)(34,44)(35,43)(39,47)(40,46)(45,48);(2,5)(6,8)(10,13)(14,15)(18,24)(19,29)(22,35)(23,36)(25,39)(26,27)(28,38)(30,42)(32,40)(34,45)(37,46)(41,47);(1,3,4,7)(2,6,5,8)(17,20,21,33)(18,25,26,32)(19,30,29,42)(22,34,35,45)(23,37,28,41)(24,39,27,40)(31,44,43,48)(36,46,38,47);(9,11,12,16)(10,14,13,15)(20,33)(25,32)(30,42)(34,45)(37,41)(39,40)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6708004);