#499762

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
prime radical	#61053	C₂
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:

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

Polycyclic presentation:

2^9|a^2=i,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,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, 406958);