#7613474

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563341
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1874 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,24)(13,18)(14,17)(15,22)(16,28)(19,26)(21,37)(23,32)(25,36)(27,34)(29,31)(30,40)(33,39)(35,38);(2,5)(6,8)(9,11,13,22)(10,15,18,20)(12,25,27,40)(14,29,26,23)(16,33,35,37)(17,32,19,31)(21,38,39,28)(24,30,34,36);(1,3,4,7)(2,6,5,8)(9,12,26,38,13,27,14,28)(10,16,17,34,18,35,19,24)(11,21,29,40,22,39,23,25)(15,30,31,37,20,36,32,33);(10,17,18,19)(15,31,20,32)(16,34,35,24)(30,37,36,33);(2,5)(6,8)(10,18)(12,27)(15,20)(17,19)(21,39)(25,40)(28,38)(31,32)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=h,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,g^b=gi,e^c=ei

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, 7520670);