#6533902

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 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)(17,18,23,21)(19,24,22,31)(20,27,30,32)(25,28,29,26);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,21,31)(20,28,30,26)(25,27,29,32);(10,13)(14,16)(17,20)(18,25)(19,26)(21,29)(22,28)(23,30)(24,32)(27,31);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,18)(19,24,22,31)(20,29,30,25)(26,32,28,27);(11,15)(14,16)(17,19,23,22)(18,24,21,31)(20,26,30,28)(25,32,29,27)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6441098);