#7624023

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623604
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 ◁ #529 ◁ #2966 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322566
maximal normal subgroup of	#72322566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531219);