#6894212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#37296	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,7,8,5)(17,18,21,26,22,27,30,32)(19,25,29,31,20,23,28,24);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,22)(21,29)(24,26)(25,27)(28,30)(31,32);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,22)(21,28)(23,26)(25,32)(27,31)(29,30);(2,6)(5,7)(11,16)(13,15)(17,20,22,19)(18,25,27,23)(21,28,30,29)(24,26,31,32);(10,14)(13,15)(17,21,22,30)(18,26,27,32)(19,28,20,29)(23,31,25,24)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6801408);