#6802704

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667700
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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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)(3,7)(4,6)(5,8)(17,18,21,26,22,24,20,25)(19,27,28,31,29,32,30,23);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14);(3,8)(5,7)(10,13)(14,15)(17,19,22,29)(18,23,24,31)(20,28,21,30)(25,27,26,32);(2,6)(5,7)(18,24)(23,31)(25,26)(27,32);(9,11,12,16)(10,14,13,15)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi

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, 6709900);