#6989592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1527 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)(9,10,14,13)(11,15,12,16)(17,18)(19,21)(20,27)(22,26)(23,29)(24,30)(25,31)(28,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,19,21,18)(20,23,28,25)(22,26,30,24)(27,31,32,29);(3,8)(5,7)(9,12,14,11)(10,15,13,16)(17,20,22,29,21,28,30,31)(18,23,24,27,19,25,26,32);(2,6)(5,7)(9,10,14,13)(11,15,12,16)(18,19)(20,28)(24,26)(29,31);(9,13,14,10)(11,15,12,16)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6896788);