#7613465

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

1916 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)(11,21)(12,25)(13,20)(14,19)(15,18)(16,24)(17,29)(22,38)(23,33)(26,37)(27,36)(28,35)(30,32)(31,40)(34,39);(2,5)(6,8)(9,11,14,24)(10,16,19,21)(12,26,28,40)(13,23,15,30)(17,34,36,38)(18,33,20,32)(22,27,39,29)(25,31,35,37);(1,3,4,7)(2,6,5,8)(9,12,13,27,14,28,15,29)(10,17,18,35,19,36,20,25)(11,22,30,40,24,39,23,26)(16,31,32,38,21,37,33,34);(10,18,19,20)(16,32,21,33)(17,35,36,25)(31,38,37,34);(2,5)(6,8)(9,13,14,15)(10,18,19,20)(11,23,24,30)(12,27,28,29)(16,33,21,32)(17,35,36,25)(22,26,39,40)(31,34,37,38)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=h,e^2=i,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, 7520661);