#7501137

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

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

2640 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,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, 7408333);