#6687395

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

1691 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18)(19,28)(20,34)(21,27)(22,26)(23,32)(24,39)(25,36)(29,41)(30,31)(33,44)(35,45)(37,43)(38,42)(40,46)(47,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,23)(20,35)(21,30)(22,32)(24,41)(25,31)(26,28)(27,44)(29,37)(33,36)(34,40)(38,46)(39,48)(42,45)(43,47);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,36,37,22,38,21,39)(18,24,27,42,26,43,25,34)(19,29,33,46,32,48,30,35)(23,40,44,41,28,45,31,47);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(17,21,22,36)(18,25,26,27)(19,30,32,33)(20,37,38,39)(23,31,28,44)(24,42,43,34)(29,46,48,35)(40,41,45,47);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,18)(19,31)(20,24)(21,25)(22,26)(23,33)(27,36)(28,30)(29,47)(32,44)(34,39)(35,45)(37,42)(38,43)(40,46)(41,48)

Polycyclic presentation:

2^9|c^2=fg,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, 6594591);