#6687399

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

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

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,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, 6594595);