#6800812

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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=i,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, 6708008);