#6542810

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
prime radical	#29607	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1586 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ei,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, 6450006);