#6432088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431723
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23407	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3338 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)(11,15)(12,17)(13,16)(14,19)(18,26)(20,28)(21,27)(22,30)(23,32)(24,31)(25,29)(33,36)(34,39)(35,38)(37,40);(2,5)(6,8)(9,11,14,22)(10,15,19,30)(12,20,23,34)(13,25,26,21)(16,29,18,27)(17,28,32,39)(24,37,36,35)(31,40,33,38);(10,16)(13,26)(15,27)(17,31)(18,19)(21,25)(24,36)(28,38)(29,30)(32,33)(35,37)(39,40);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,24)(14,23)(15,28)(16,31)(18,33)(19,32)(21,35)(22,34)(25,37)(26,36)(27,38)(29,40)(30,39);(12,23)(17,32)(20,34)(24,36)(28,39)(31,33)(35,37)(38,40)

Polycyclic presentation:

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

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, 6339284);