#6514382

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60564	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1736 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6421578);