#6687405

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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)(21,25)(22,27)(23,31)(24,38)(26,36)(29,41)(30,32)(33,43)(35,45)(37,44)(39,42)(40,48)(46,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,19)(18,23)(20,35)(21,31)(22,32)(24,41)(25,28)(26,30)(27,43)(29,39)(33,36)(34,40)(37,48)(38,46)(42,47)(44,45);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,36,39,21,37,22,38)(18,24,27,44,25,42,26,34)(19,29,33,48,31,46,32,35)(23,40,43,41,28,45,30,47);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(18,25)(20,37)(23,28)(26,27)(29,46)(30,43)(35,48)(38,39);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,18)(19,30)(20,24)(21,25)(22,26)(23,33)(27,36)(28,32)(29,47)(31,43)(34,38)(35,45)(37,42)(39,44)(40,48)(41,46)

Polycyclic presentation:

2^9|c^2=fg,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,d^c=di,h^c=hi,e^d=ei,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, 6594601);