#7623282

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622982
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 ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1370 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)(11,17)(12,25)(13,18)(14,22)(15,34)(16,21)(19,27)(20,23)(24,35)(26,33)(28,38)(29,39)(30,40)(31,37)(32,36);(2,5)(6,8)(9,11,16,24)(10,17,21,35)(12,26,29,22)(13,23,31,15)(14,25,33,39)(18,20,37,34)(19,36,28,30)(27,32,38,40);(9,12,27,37,15,30,33,35,16,29,38,18,23,36,14,17)(10,11,22,32,20,13,28,39,21,24,26,40,34,31,19,25);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,23)(12,28)(14,32)(15,24)(16,31)(17,20)(19,29)(21,37)(22,36)(25,38)(26,30)(27,39)(33,40)(34,35);(11,24)(12,29)(13,31)(17,35)(18,37)(25,39)(30,36)(32,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322758
maximal normal subgroup of	#72322758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530478);