#7623322

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623311
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1633 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530518);