#7503964

order	512
nilpotency class	3
isoclinism	isoclinic class of #7501009
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

1788 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)(3,4)(5,7)(6,8)(9,10,15,30,14,19,13,20)(11,16,23,33,24,32,22,18)(12,25,29,17,28,36,27,35)(21,37,26,31,39,34,38,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,26)(13,22)(14,24)(15,23)(17,34)(18,20)(19,32)(21,27)(25,31)(28,38)(29,39)(30,33)(35,37)(36,40);(9,12,14,28)(10,17,19,35)(11,21,24,39)(13,29,15,27)(16,31,32,40)(18,34,33,37)(20,36,30,25)(22,26,23,38);(2,7)(3,8)(9,11)(10,18)(12,21)(13,23)(14,24)(15,22)(16,20)(17,34)(19,33)(25,40)(26,27)(28,39)(29,38)(30,32)(31,36)(35,37);(1,4,5,6)(2,3,7,8)(10,19)(16,32)(17,35)(18,33)(20,30)(25,36)(31,40)(34,37)

Polycyclic presentation:

2^9|a^2=gi,c^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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, 7411160);