#7624721

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

1209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18,22,29)(19,31,20,25)(21,27,24,28)(23,38,37,41)(26,33,46,35)(30,36,47,40)(32,43,34,42)(39,44,48,45);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,22,20)(18,25,29,31)(21,34,24,32)(23,33,37,35)(26,41,46,38)(27,43,28,42)(30,44,47,45)(36,39,40,48);(17,20,22,19)(18,26,29,46)(21,34,24,32)(23,39,37,48)(25,41,31,38)(27,44,28,45)(30,43,47,42)(33,40,35,36);(3,10)(6,9)(11,16)(14,15)(17,21,22,24)(18,27,29,28)(19,32,20,34)(23,36,37,40)(25,42,31,43)(26,44,46,45)(30,38,47,41)(33,48,35,39);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,21,22,24)(18,28,29,27)(19,32,20,34)(23,36,37,40)(25,43,31,42)(26,45,46,44)(30,41,47,38)(33,48,35,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,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, 7531917);