#499768

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
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

1254 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)(7,21)(8,26)(9,31)(10,33)(11,19)(12,18)(13,20)(14,23)(15,29)(16,32)(17,37)(22,53)(24,46)(25,52)(27,51)(28,47)(30,50)(34,44)(35,42)(36,49)(38,48)(39,45)(40,64)(41,57)(43,54)(55,63)(56,68)(58,67)(59,61)(60,62)(65,66);(1,2)(3,4)(5,7,13,25)(6,14,20,42)(8,22,30,55)(9,12,24,11)(10,32,38,57)(15,40,45,59)(16,18,41,19)(17,46,49,31)(21,48,52,33)(23,36,35,37)(26,43,50,68)(27,29,58,39)(28,61,34,64)(44,63,47,53)(51,60,67,65)(54,66,56,62);(2,4)(5,8)(6,15)(7,22)(9,27)(10,34)(11,39)(12,29)(13,30)(14,40)(16,43)(17,47)(18,50)(19,26)(20,45)(21,51)(23,56)(24,58)(25,55)(28,38)(31,63)(32,64)(33,65)(35,54)(36,62)(37,66)(41,68)(42,59)(44,49)(46,53)(48,60)(52,67)(57,61);(5,9,13,24)(6,16,20,41)(7,11,25,12)(8,27,30,58)(10,35,38,23)(14,19,42,18)(15,43,45,68)(17,21,49,52)(22,39,55,29)(26,59,50,40)(28,56,34,54)(31,48,46,33)(32,36,57,37)(44,67,47,51)(53,65,63,60)(61,66,64,62)

Polycyclic presentation:

2^9|b^2=i,d^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,d^b=di,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406964);