#7623736

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623613
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 ◁ #525 ◁ #2929 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1530 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)(11,21)(12,14)(13,25)(16,29)(17,20)(19,34)(22,38)(23,37)(24,27)(26,28)(30,40)(31,39)(32,33)(35,36);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,17)(13,19)(14,20)(15,18)(21,29)(22,30)(23,31)(24,32)(25,34)(26,35)(27,33)(28,36)(37,39)(38,40);(9,11)(10,16)(12,22)(13,26)(14,28)(15,23)(17,30)(18,31)(19,35)(20,36)(21,25)(24,37)(27,38)(29,34)(32,39)(33,40);(3,8)(5,7)(9,12,15,14)(10,17,18,20)(11,22,23,28)(13,24,27,25)(16,30,31,36)(19,32,33,34)(21,26,37,38)(29,35,39,40);(10,18)(16,31)(17,20)(19,33)(29,39)(30,36)(32,34)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322060
maximal normal subgroup of	#72322060
maximal quotient of	#72343855, #72343858

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530932);