#6675873

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

1668 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)(19,31)(20,36)(21,30)(22,28)(23,29)(24,27)(25,34)(26,38)(32,46)(33,43)(35,45)(37,48)(39,44)(40,47)(41,42);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,25)(20,37)(21,33)(22,38)(23,34)(24,35)(26,46)(27,43)(28,47)(29,31)(30,45)(32,40)(36,42)(39,41)(44,48);(17,20,23,39)(18,26,29,47)(19,32,34,22)(21,38,24,40)(25,42,31,48)(27,36,30,44)(28,45,46,43)(33,37,35,41);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,24)(18,27,29,30)(19,33,34,35)(20,38,39,40)(22,41,32,37)(25,43,31,45)(26,36,47,44)(28,48,46,42);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,22)(18,28)(19,20)(21,37)(23,32)(24,41)(25,44)(26,45)(27,42)(29,46)(30,48)(31,36)(33,40)(34,39)(35,38)(43,47)

Polycyclic presentation:

2^9|c^2=i,d^2=f,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, 6583069);