#7623049

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622820
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
prime radical	C₂×#3061	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1604 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)(3,6)(4,5)(7,8)(9,10)(11,21)(12,17)(13,26)(14,18)(15,20)(16,23)(19,28)(22,37)(24,38)(25,36)(27,31)(29,34)(30,35)(32,40)(33,39);(2,5)(6,8)(9,10)(11,16)(12,18)(13,19)(14,17)(15,20)(21,23)(22,32)(24,33)(25,34)(26,28)(27,35)(29,36)(30,31)(37,40)(38,39);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,27)(14,30)(15,24)(17,31)(18,32)(19,35)(20,33)(21,26)(23,28)(25,39)(29,40)(34,38)(36,37);(9,12,15,14)(10,17,20,18)(11,22,24,30)(13,25,29,28)(16,31,33,32)(19,26,36,34)(21,37,38,35)(23,27,39,40);(9,12,15,14)(10,18,20,17)(11,22,24,30)(13,25,29,28)(16,32,33,31)(19,34,36,26)(21,35,38,37)(23,27,39,40)

Polycyclic presentation:

2^9|c^2=f,d^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gh,h^a=hi,d^b=di,g^c=gh,h^c=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, 7530245);