#7624570

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624150
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

1328 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)(3,7)(9,10)(11,22)(12,18)(13,20)(14,29)(15,19)(16,24)(17,25)(21,31)(23,34)(26,33)(27,38)(28,32)(30,36)(35,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,26)(13,24)(14,25)(15,23)(17,32)(18,34)(19,33)(20,22)(21,35)(27,40)(28,39)(29,37)(30,31)(36,38);(9,11)(10,17)(12,23)(13,24)(14,30)(15,26)(16,32)(18,36)(19,35)(20,37)(21,33)(22,29)(25,31)(27,39)(28,40)(34,38);(2,5)(6,8)(9,12,13,15)(10,18,20,19)(11,23,24,26)(14,27,28,31)(16,33,22,34)(17,36,37,35)(21,29,38,32)(25,30,39,40);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,19,20,18)(11,23,24,26)(14,27,28,31)(16,34,22,33)(17,35,37,36)(21,32,38,29)(25,30,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531766);