#6502938

order	512
nilpotency class	3
isoclinism	isoclinic class of #6502937
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1994 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,f^2=i,g^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, 6410134);