#6675534

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

1646 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)(19,24)(20,32)(21,26)(22,27)(23,29)(25,35)(28,37)(30,41)(31,33)(34,46)(36,42)(38,43)(39,44)(40,45)(47,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,29)(21,25)(22,38)(23,32)(26,35)(27,43)(28,33)(30,39)(31,37)(34,42)(36,46)(40,47)(41,44)(45,48);(17,20,23,24)(18,25,29,31)(19,26,32,28)(21,33,37,35)(22,34,40,41)(27,38,45,47)(30,42,46,44)(36,48,39,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,23,37)(18,26,29,28)(19,25,32,31)(20,33,24,35)(22,36,40,39)(27,42,45,44)(30,38,46,47)(34,48,41,43);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,22)(18,27)(19,30)(20,34)(21,36)(23,40)(24,41)(25,38)(26,42)(28,44)(29,45)(31,47)(32,46)(33,48)(35,43)(37,39)

Polycyclic presentation:

2^9|c^2=i,d^2=fi,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, 6582730);