#7501819

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2323 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)(3,5)(4,6)(7,8)(9,10)(11,21)(12,23)(13,28)(14,18)(15,19)(16,27)(17,30)(20,33)(22,38)(24,26)(25,39)(29,40)(31,36)(32,37)(34,35);(1,3,5,2)(4,7,8,6)(9,11,15,23)(10,12,19,21)(13,29,31,22)(14,16,33,24)(17,35,37,25)(18,26,20,27)(28,38,36,40)(30,39,32,34);(1,4,5,8)(2,6,3,7)(9,12,15,21)(10,16,19,24)(11,18,23,20)(13,25,31,35)(14,26,33,27)(17,34,37,39)(22,36,29,28)(30,38,32,40);(2,3)(6,7)(9,13)(10,17)(11,22)(12,25)(14,30)(15,31)(16,34)(18,36)(19,37)(20,28)(21,35)(23,29)(24,39)(26,38)(27,40)(32,33);(9,14,15,33)(10,18,19,20)(11,16,23,24)(12,26,21,27)(13,30,31,32)(17,36,37,28)(22,34,29,39)(25,38,35,40)

Polycyclic presentation:

2^9|b^2=f,c^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, 7409015);