#6463388

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1545 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18,22,27)(19,28,25,24)(20,30,31,29)(21,26,23,32);(2,5)(6,8)(11,12)(14,16)(17,19)(18,24)(20,26)(21,30)(22,25)(23,29)(27,28)(31,32);(9,11,13,12)(10,14,15,16)(17,20,21,24,22,31,23,28)(18,25,32,29,27,19,26,30);(9,12,13,11)(10,14,15,16)(17,21,22,23)(18,26,27,32)(19,29,25,30)(20,24,31,28);(1,3,4,7)(2,6,5,8)(19,25)(20,31)(24,28)(29,30)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370584);