#6432089

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	#62855	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3410 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)(11,19)(12,16)(13,25)(14,18)(15,21)(17,26)(20,33)(22,32)(23,30)(24,37)(27,28)(29,35)(31,36)(34,40)(38,39);(2,5)(6,8)(13,26)(15,28)(17,25)(21,27)(24,36)(29,39)(31,37)(35,38);(9,11,14,22)(10,15,18,28)(12,20,23,34)(13,27,26,21)(16,29,30,39)(17,32,25,19)(24,38,36,35)(31,40,37,33);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,20)(13,24)(14,23)(15,29)(17,31)(18,30)(19,33)(21,35)(22,34)(25,37)(26,36)(27,38)(28,39)(32,40);(12,23)(16,30)(20,34)(24,36)(29,39)(31,37)(33,40)(35,38)

Polycyclic presentation:

2^9|c^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.

Relationships

maximal subgroup of	#72318196
maximal normal subgroup of	#72318196

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339285);