#7624283

order	512
nilpotency class	4
isoclinism	isoclinic class of #7624128
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

1247 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)(19,28)(20,26)(21,33)(22,37)(23,25)(24,31)(27,35)(29,44)(30,45)(32,42)(34,40)(36,47)(38,43)(39,46)(41,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(19,29)(24,39)(28,44)(30,38)(31,46)(34,48)(40,41)(43,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,24)(20,30)(21,34)(22,38)(23,29)(25,39)(26,40)(27,43)(28,33)(31,35)(32,46)(36,48)(37,41)(42,45)(44,47);(3,11)(6,9)(10,16)(13,15)(18,25)(24,39)(26,37)(27,42)(28,44)(33,47)(40,41)(43,45);(2,8)(6,9)(7,14)(13,15)(17,20,23,22)(18,26,25,37)(19,30,29,38)(21,32,36,35)(24,40,39,41)(27,33,42,47)(28,45,44,43)(31,34,46,48)

Polycyclic presentation:

2^9|c^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,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, 7531479);