#7623724

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×#509
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	D₄×#135	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1538 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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=df,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, 7530920);