#499767

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
prime radical	#61054	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1260 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:

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

Polycyclic presentation:

2^9|a^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, 406963);