#6513156

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2984 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,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, 6420352);