#509172

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

767 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416368);