#6508596

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6415792);