#7593013

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

1696 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)(11,23)(12,29)(13,18)(14,21)(15,20)(16,22)(17,26)(19,32)(24,33)(25,34)(27,36)(28,35)(30,39)(31,38)(37,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,26,16,28,15,27)(10,17,20,23,22,36,21,35)(12,30,13,24,32,40,31,25)(18,37,29,33,38,39,19,34);(9,12,15,31,16,32,14,13)(10,18,21,19,22,38,20,29)(11,24,26,40,28,25,27,30)(17,33,23,39,36,34,35,37);(3,7)(5,8)(9,12,15,31,16,32,14,13)(10,18,21,19,22,38,20,29)(11,25,26,30,28,24,27,40)(17,34,23,37,36,33,35,39);(9,13,14,32,16,31,15,12)(10,19,20,18,22,29,21,38)(11,24,26,40,28,25,27,30)(17,34,23,37,36,33,35,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=gi,d^2=gi,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, 7500209);