#7613463

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563319
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

1959 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)(23,31)(25,36)(27,33)(29,35)(30,39)(32,38)(34,40);(2,5)(6,8)(9,11)(10,15)(12,25)(13,22)(14,29)(16,32)(17,31)(18,20)(19,35)(21,28)(23,26)(24,30)(27,39)(33,36)(34,37)(38,40);(1,3,4,7)(2,6,5,8)(9,12,26,40,13,27,14,28)(10,16,17,33,18,34,19,24)(11,21,29,39,22,38,23,25)(15,30,35,37,20,36,31,32);(10,17,18,19)(11,22)(15,31,20,35)(16,33,34,24)(21,38)(23,29)(25,39)(30,32,36,37);(2,5)(6,8)(10,18)(15,20)(16,34)(17,19)(24,33)(30,36)(31,35)(32,37)

Polycyclic presentation:

2^9|c^2=fg,d^2=hi,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520659);