#6491476

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
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

2945 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)(9,10,15,20,16,21,14,19)(11,22,25,32,26,35,30,17)(12,27,28,34,13,18,29,33)(23,36,39,31,24,37,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,23)(13,24)(14,25)(15,30)(16,26)(18,31)(19,22)(20,35)(21,32)(27,40)(28,38)(29,39)(33,36)(34,37);(2,5)(6,8)(9,12,16,13)(10,18,21,27)(11,23,26,24)(14,28,15,29)(17,31,32,40)(19,33,20,34)(22,36,35,37)(25,38,30,39);(2,5)(6,8)(9,13,16,12)(10,18,21,27)(11,24,26,23)(14,28,15,29)(17,31,32,40)(19,34,20,33)(22,37,35,36)(25,38,30,39);(1,3,4,7)(2,6,5,8)(9,14,16,15)(10,19,21,20)(11,25,26,30)(12,28,13,29)(17,22,32,35)(18,33,27,34)(23,38,24,39)(31,36,40,37)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6398672);