#502513

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	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 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)(5,6,13,19)(7,20,25,52)(8,15,30,51)(9,31,36,64)(10,37,39,66)(11,18,12,42)(14,43,48,23)(16,22,54,33)(17,38,55,21)(24,57,40,58)(26,62,61,53)(27,46,32,47)(28,56,60,67)(29,50,41,45)(34,63,35,44)(49,65,68,59);(1,3,4,2)(5,7,21,27,12,24,39,43,13,25,38,32,11,40,10,23)(6,14,37,58,18,47,55,52,19,48,66,57,42,46,17,20)(8,26,59,35,29,22,60,36,30,61,65,34,41,33,28,9)(15,31,56,16,45,63,68,53,51,64,67,54,50,44,49,62);(2,3)(5,8)(6,15)(7,22)(9,32)(10,28)(11,41)(12,29)(13,30)(14,44)(16,52)(17,49)(18,45)(19,51)(20,54)(21,59)(23,35)(24,61)(25,33)(26,40)(27,36)(31,47)(34,43)(37,56)(38,65)(39,60)(42,50)(46,64)(48,63)(53,57)(55,68)(58,62)(66,67);(5,9,21,26,12,35,39,22,13,36,38,61,11,34,10,33)(6,16,37,63,18,53,55,64,19,54,66,44,42,62,17,31)(7,8,27,59,24,29,43,60,25,30,32,65,40,41,23,28)(14,45,58,68,47,51,52,67,48,50,57,49,46,15,20,56)

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,d^2=eh,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, 409709);