#6686587

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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

1601 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)(19,29)(20,35)(21,28)(22,27)(23,26)(24,33)(25,37)(30,46)(31,32)(34,41)(36,40)(38,44)(39,43)(42,48)(45,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,36)(21,31)(22,33)(23,34)(25,42)(26,32)(27,29)(28,41)(30,37)(35,45)(38,47)(39,48)(40,44)(43,46);(17,20,23,37,22,38,21,39)(18,25,28,35,27,43,26,44)(19,30,34,47,33,48,31,36)(24,40,41,42,29,45,32,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(17,21,22,23)(18,26,27,28)(19,31,33,34)(20,37,38,39)(24,32,29,41)(25,35,43,44)(30,47,48,36)(40,42,45,46);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,18)(19,32)(20,25)(21,26)(22,27)(23,28)(24,34)(29,31)(30,46)(33,41)(35,37)(36,45)(38,43)(39,44)(40,47)(42,48)

Polycyclic presentation:

2^9|c^2=g,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,d^c=di,h^c=hi,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, 6593783);