#502515

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×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

650 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,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, 409711);