#498806

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#466
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1490 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406002);