#6432082

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318727, #72318775, #72318940, …
maximal normal subgroup of	#72318727, #72318775, #72318940, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339278);