#6687403

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

1712 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)(20,25)(21,36)(22,26)(23,27)(24,32)(29,30)(31,43)(33,35)(34,41)(37,39)(38,42)(40,46)(44,45)(47,48);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,16)(10,11)(17,18)(19,29)(20,33)(21,26)(22,36)(23,27)(24,32)(25,35)(28,30)(31,40)(34,42)(37,39)(38,41)(43,46)(44,47)(45,48);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19,21,32,23,30,22,37)(18,24,26,28,27,39,36,29)(20,31,35,46,34,45,38,47)(25,40,42,43,41,48,33,44);(1,4,6,5)(2,9,10,8)(3,12,13,14)(7,16,11,15)(19,30)(20,34)(24,39)(25,41)(28,29)(32,37)(33,42)(35,38);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,20,23,34)(18,25,27,41)(19,31,30,45)(21,35,22,38)(24,40,39,48)(26,42,36,33)(28,43,29,44)(32,46,37,47)

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,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, 6594599);