#7624022

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

1329 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:

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

Polycyclic presentation:

2^9|a^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	#72260392, #72322566
maximal normal subgroup of	#72260392, #72322566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531218);