#6675883

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667964
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 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)(3,10)(4,13)(5,8)(6,9)(7,12)(11,15)(14,16)(17,18)(19,28)(20,33)(21,26)(22,38)(23,27)(24,31)(25,36)(29,42)(30,39)(32,41)(34,45)(35,44)(37,43)(40,48)(46,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19,21,31)(18,24,26,28)(20,34,35,48)(22,32,23,39)(25,42,43,46)(27,41,38,30)(29,36,47,37)(33,40,44,45);(17,20,21,35)(18,25,26,43)(19,29,31,47)(22,37,23,36)(24,40,28,45)(27,44,38,33)(30,46,41,42)(32,48,39,34);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,26)(24,28)(25,43)(27,38)(30,41)(33,44)(40,45)(42,46);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,18)(19,30)(20,25)(21,26)(22,27)(23,38)(24,32)(28,39)(29,46)(31,41)(33,36)(34,45)(35,43)(37,44)(40,48)(42,47)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6583079);