#7500735

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2712 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:

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

Polycyclic presentation:

2^9|e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,h^b=hi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7407931);