#6507366

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6414562);