#7624366

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624235
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 ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=f,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,d^b=dfi,g^c=gh,h^c=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531562);