#511892

order	512
nilpotency class	4
isoclinism	isoclinic class of #511890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6,12,18)(7,19,21,38)(8,25,28,58)(9,17,11,15)(10,30,33,42)(13,24,36,22)(14,39,41,27)(16,29,44,32)(20,48,49,68)(23,37,52,64)(26,59,34,43)(31,45,61,40)(35,62,63,54)(46,66,65,51)(47,53,60,55)(50,57,56,67);(5,7,11,24,12,21,9,22)(6,13,17,38,18,36,15,19)(8,20,34,56,28,49,26,50)(10,23,32,55,33,52,29,53)(14,35,45,65,41,63,40,46)(16,37,30,60,44,64,42,47)(25,57,59,68,58,67,43,48)(27,51,61,62,39,66,31,54);(5,8)(6,14)(7,20)(9,26)(10,31)(11,34)(12,28)(13,35)(15,40)(16,43)(17,45)(18,41)(19,46)(21,49)(22,50)(23,54)(24,56)(25,30)(27,32)(29,39)(33,61)(36,63)(37,48)(38,65)(42,58)(44,59)(47,67)(51,55)(52,62)(53,66)(57,60)(64,68);(2,4)(7,21)(13,36)(19,38)(20,49)(22,24)(23,52)(35,63)(37,64)(46,65)(47,60)(48,68)(50,56)(51,66)(53,55)(54,62)(57,67)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fh,h^c=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, 419088);