#7624218

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624184
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 ◁ #532 ◁ #2931 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1521 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)(3,7)(4,6)(5,8)(10,17)(11,21)(13,16)(15,29)(18,33)(20,26)(22,35)(23,37)(24,39)(25,38)(27,28)(30,31)(32,36)(34,40);(1,3)(2,5)(4,8)(6,7)(9,10,14,17)(11,22,25,18)(12,16,19,13)(15,20,28,32)(21,33,38,35)(23,24,34,31)(26,29,36,27)(30,40,39,37);(9,11)(10,18)(12,23)(13,24)(14,25)(15,30)(16,31)(17,22)(19,34)(20,37)(21,29)(26,33)(27,38)(28,39)(32,40)(35,36);(3,8)(5,7)(9,12,14,19)(10,13,17,16)(11,23,25,34)(15,26,28,36)(18,24,22,31)(20,27,32,29)(21,37,38,40)(30,33,39,35);(2,6)(5,7)(9,13,14,16)(10,19,17,12)(11,24,25,31)(15,27,28,29)(18,34,22,23)(20,36,32,26)(21,30,38,39)(33,37,35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531414);