#7624678

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 ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1249 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:

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

Polycyclic presentation:

2^9|a^2=f,e^2=fi,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,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287767, #72320055, #72320093, …
maximal normal subgroup of	#72287767, #72320055, #72320093, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7531874);