#499759

order	512
nilpotency class	4
isoclinism	isoclinic class of #499757
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1276 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)(7,20)(8,26)(9,15)(10,18)(11,17)(12,19)(13,22)(14,28)(16,32)(21,51)(23,45)(24,49)(25,50)(27,46)(29,47)(30,37)(31,41)(33,43)(34,44)(35,42)(36,63)(38,56)(39,57)(40,64)(48,53)(52,59)(54,68)(55,66)(58,65)(60,62)(61,67);(1,2)(3,4)(5,7,12,25)(6,13,19,39)(8,21,29,54)(9,30,34,56)(10,24,11,23)(14,36,42,67)(15,20,44,50)(16,45,43,49)(17,38,18,37)(22,33,57,32)(26,59,47,66)(27,61,31,63)(28,58,35,53)(40,48,60,65)(41,68,46,51)(52,64,55,62);(2,4)(5,8)(6,14)(7,21)(9,31)(10,35)(11,28)(12,29)(13,36)(15,40)(16,46)(17,47)(18,26)(19,42)(20,48)(22,55)(23,58)(24,53)(25,54)(27,34)(30,63)(32,64)(33,62)(37,59)(38,66)(39,67)(41,43)(44,60)(45,51)(49,68)(50,65)(52,57)(56,61);(6,15)(9,32)(10,11)(13,20)(14,40)(16,18)(17,43)(19,44)(22,30)(23,24)(26,46)(27,62)(28,35)(31,64)(33,34)(36,48)(37,45)(38,49)(39,50)(41,47)(42,60)(51,59)(52,61)(53,58)(55,63)(56,57)(65,67)(66,68)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406955);