#6675535

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

1666 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)(19,30)(20,34)(21,28)(22,27)(23,40)(24,29)(25,32)(26,37)(31,46)(33,44)(35,48)(36,43)(38,47)(39,42)(41,45);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,25)(20,35)(21,37)(22,32)(23,33)(24,41)(26,46)(27,30)(28,47)(29,44)(31,38)(34,42)(36,39)(40,45)(43,48);(17,20,22,36)(18,26,27,47)(19,31,32,21)(23,38,24,37)(25,42,30,48)(28,44,46,45)(29,43,40,34)(33,39,41,35);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,27)(25,30)(26,47)(28,46)(29,40)(34,43)(42,48)(44,45);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21)(18,28)(19,20)(22,31)(23,39)(24,35)(25,43)(26,44)(27,46)(29,48)(30,34)(32,36)(33,37)(38,41)(40,42)(45,47)

Polycyclic presentation:

2^9|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,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, 6582731);