#6990132

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	#29806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1507 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,21)(20,27)(22,26)(23,29)(24,30)(25,31)(28,32);(1,2,6,3)(4,7,5,8)(9,11,12,16)(10,13,14,15)(17,19,21,18)(20,23,28,25)(22,26,30,24)(27,31,32,29);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,20,22,29,21,28,30,31)(18,23,24,27,19,25,26,32);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(18,19)(20,28)(24,26)(29,31);(1,5,6,4)(2,7,3,8)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,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, 6897328);