#501694

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

602 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)(7,20)(8,27)(9,34)(10,39)(11,17)(12,18)(13,23)(14,31)(15,22)(16,21)(19,40)(24,49)(25,57)(26,63)(28,52)(29,45)(30,56)(32,43)(33,48)(35,55)(36,54)(37,46)(38,42)(41,50)(44,61)(47,65)(51,60)(53,62)(58,64)(59,67)(66,68);(5,7,21,35,12,26,41,61,11,24,36,29,40,25,10,23)(6,13,39,57,19,45,54,49,17,44,50,63,18,55,16,20)(8,28,58,38,33,60,66,37,31,62,65,59,32,22,30,9)(14,46,68,51,48,42,64,52,27,34,56,15,43,67,47,53);(2,4)(5,8)(6,14)(7,22)(9,35)(10,30)(11,31)(12,33)(13,42)(15,49)(16,47)(17,27)(18,43)(19,48)(20,51)(21,58)(23,59)(24,60)(25,62)(26,28)(29,37)(32,40)(34,45)(36,65)(38,61)(39,68)(41,66)(44,67)(46,55)(50,56)(52,57)(53,63)(54,64);(5,9,36,62,12,38,10,22,11,37,21,28,40,59,41,60)(6,15,50,42,19,53,16,34,17,51,39,67,18,52,54,46)(7,8,29,65,26,33,23,30,24,31,35,58,25,32,61,66)(13,43,63,64,45,14,20,56,44,48,57,47,55,27,49,68)

Polycyclic presentation:

2^9|a^2=fi,b^2=eh,d^2=ehi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

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, 408890);