#6435414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432881
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	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2861 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=i,d^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,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, 6342610);