#6502311

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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	#24210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 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)(3,8)(4,6)(5,7)(17,18)(19,23)(20,25)(21,24)(22,27)(26,30)(28,32)(29,31);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,20)(18,23,27,25)(21,28,30,29)(24,32,26,31);(9,11)(10,13)(12,16)(14,15)(18,24)(19,20)(21,30)(23,31)(25,32)(26,27);(11,16)(13,15)(17,20,22,19)(18,25,27,23)(21,29,30,28)(24,31,26,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,20)(23,25)(28,29)(31,32)

Polycyclic presentation:

2^9|b^2=fi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,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, 6409507);