#6687787

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
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1625 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,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,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, 6594983);