#6675530

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667748
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)(17,18,23,22)(19,28,32,24)(20,34,38,44)(21,27,40,26)(25,36,37,39)(29,46,47,43)(30,42,33,31)(35,41,48,45);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,35)(21,30)(22,28)(23,32)(25,43)(26,42)(27,31)(29,39)(33,40)(34,45)(36,47)(37,46)(38,48)(41,44);(17,20,23,38)(18,25,22,37)(19,29,32,47)(21,36,40,39)(24,41,28,45)(26,44,27,34)(30,35,33,48)(31,43,42,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,40)(18,26,22,27)(19,30,32,33)(20,36,38,39)(24,42,28,31)(25,44,37,34)(29,35,47,48)(41,46,45,43);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,22,23,18)(19,31,32,42)(20,37,38,25)(21,27,40,26)(24,33,28,30)(29,43,47,46)(34,39,44,36)(35,41,48,45)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=f,e^2=f,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,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582726);