#7502564

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500736
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=fi,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,e^c=ei,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, 7409760);