#502517

order	512
nilpotency class	4
isoclinism	isoclinic class of #502469
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	#33610	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

648 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409713);