#7593011

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

1662 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=gi,d^2=g,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi

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, 7500207);