#499765

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	#61051	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1266 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)(7,20,22,38)(8,26,29,49)(9,15,34,46)(10,18,11,17)(13,36,41,23)(14,35,44,28)(16,33,45,32)(21,52,53,68)(24,47,25,51)(27,48,31,43)(30,39,57,40)(37,61,67,63)(42,62,60,64)(50,58,65,55)(54,59,56,66);(1,2)(3,4)(5,7,12,22)(6,13,19,41)(8,21,29,53)(9,30,34,57)(10,25,11,24)(14,37,44,67)(15,20,46,38)(16,47,45,51)(17,40,18,39)(23,33,36,32)(26,59,49,66)(27,61,31,63)(28,58,35,55)(42,50,60,65)(43,68,48,52)(54,64,56,62);(2,4)(5,8)(6,14)(7,21)(9,31)(10,35)(11,28)(12,29)(13,37)(15,42)(16,48)(17,49)(18,26)(19,44)(20,50)(22,53)(23,56)(24,58)(25,55)(27,34)(30,63)(32,64)(33,62)(36,54)(38,65)(39,59)(40,66)(41,67)(43,45)(46,60)(47,52)(51,68)(57,61);(6,15)(7,22)(9,32)(10,11)(13,38)(14,42)(16,18)(17,45)(19,46)(20,41)(21,53)(23,57)(26,48)(27,62)(28,35)(30,36)(31,64)(33,34)(37,65)(39,51)(40,47)(43,49)(44,60)(50,67)(52,66)(54,63)(56,61)(59,68)

Polycyclic presentation:

2^9|a^2=i,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,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, 406961);