#6445188

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,20)(11,18,23,17)(12,25,24,32)(13,19,27,35)(15,29,16,21)(22,33,38,31)(26,40,39,34)(28,37,30,36);(2,5)(6,8)(9,11)(10,17)(12,16)(13,26)(14,23)(15,24)(18,20)(19,34)(21,32)(22,28)(25,29)(27,39)(30,38)(31,37)(33,36)(35,40);(9,12,16,23,14,24,15,11)(10,18,29,32,20,17,21,25)(13,26,30,38,27,39,28,22)(19,33,37,34,35,31,36,40);(9,13)(10,19)(11,22)(12,26)(14,27)(15,28)(16,30)(17,31)(18,33)(20,35)(21,36)(23,38)(24,39)(25,40)(29,37)(32,34);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352384);