#7518586

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
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
prime radical	#60431	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,23)(13,25)(14,28)(15,18)(17,32)(19,20)(21,35)(22,37)(24,33)(26,31)(27,30)(29,36)(34,40)(38,39);(2,5)(6,7)(9,11,15,21)(10,16,18,35)(12,24,26,34)(13,20,27,39)(14,29,32,22)(17,37,28,36)(19,30,38,25)(23,33,31,40);(9,12,15,26)(10,13,18,27)(11,19,21,38)(14,30,32,25)(16,33,35,40)(17,31,28,23)(20,36,39,37)(22,34,29,24);(9,13,15,27)(10,12,18,26)(11,20,21,39)(14,31,32,23)(16,34,35,24)(17,30,28,25)(19,36,38,37)(22,33,29,40);(1,3,4,8)(2,6,5,7)(11,21)(16,35)(19,38)(20,39)(22,29)(24,34)(33,40)(36,37)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,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, 7425782);