#509165

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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
prime radical	C₂×#576	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

769 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)(5,6,9,15,32,50,62,42,10,16,33,51,11,17,34,52)(7,18,21,53,35,66,65,38,22,56,63,39,23,49,30,12)(8,24,26,48,28,45,64,47,27,13,40,43,29,44,31,14)(19,55,20,37,25,46,60,36,57,54,59,41,58,68,61,67);(5,7,10,22)(6,12,16,38)(8,25,27,58)(9,30,33,65)(11,35,32,23)(13,41,24,37)(14,46,47,68)(15,49,51,66)(17,53,50,39)(18,42,56,52)(19,29,57,28)(20,40,59,26)(21,62,63,34)(31,60,64,61)(36,45,67,44)(43,54,48,55);(2,4)(5,8)(6,13)(7,19)(9,26)(10,27)(11,29)(12,36)(14,42)(15,43)(16,24)(17,45)(18,54)(20,21)(22,57)(23,58)(25,35)(28,32)(30,61)(31,34)(33,40)(37,39)(38,67)(41,53)(44,50)(46,49)(47,52)(48,51)(55,56)(59,63)(60,65)(62,64)(66,68);(2,4)(5,8)(6,14)(7,20)(9,31)(10,27)(11,28)(12,37)(13,42)(15,44)(16,47)(17,48)(18,55)(19,21)(22,59)(23,60)(24,52)(25,30)(26,34)(29,32)(33,64)(35,61)(36,39)(38,41)(40,62)(43,50)(45,51)(46,49)(53,67)(54,56)(57,63)(58,65)(66,68)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416361);