#6686581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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

1596 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)(19,25)(20,33)(21,27)(22,26)(23,29)(24,31)(28,34)(30,41)(32,40)(35,48)(36,42)(37,45)(38,44)(39,46)(43,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19)(18,24)(20,34)(21,35)(22,31)(23,32)(25,26)(27,43)(28,40)(29,33)(30,37)(36,47)(38,46)(39,45)(41,44)(42,48);(17,20,23,19,22,32,34,31)(18,25,28,33,26,24,29,40)(21,35,37,30,36,47,38,46)(27,41,44,48,42,39,45,43);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,26)(24,25)(27,42)(28,29)(33,40)(39,41)(43,48)(44,45);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,21)(18,27)(19,30)(20,35)(22,36)(23,37)(24,39)(25,41)(26,42)(28,44)(29,45)(31,46)(32,47)(33,48)(34,38)(40,43)

Polycyclic presentation:

2^9|c^2=g,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,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, 6593777);