#7503259

order	512
nilpotency class	3
isoclinism	isoclinic class of #7501009
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2693 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)(9,10,15,19,16,12,14,27)(11,17,23,33,24,25,22,20)(13,28,29,26,31,34,30,18)(21,35,36,39,37,40,38,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,25)(13,21)(14,22)(15,23)(16,24)(18,32)(19,33)(20,27)(26,39)(28,35)(29,36)(30,38)(31,37)(34,40);(9,12)(10,16)(11,20)(13,26)(14,27)(15,19)(17,22)(18,31)(21,35)(23,25)(24,33)(28,29)(30,34)(32,36)(37,40)(38,39);(2,6)(3,8)(9,13)(10,18)(11,21)(12,26)(14,29)(15,30)(16,31)(17,32)(19,34)(20,35)(22,36)(23,38)(24,37)(25,39)(27,28)(33,40);(9,14,16,15)(10,19,12,27)(11,22,24,23)(13,29,31,30)(17,33,25,20)(18,34,26,28)(21,36,37,38)(32,40,39,35)

Polycyclic presentation:

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

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, 7410455);