#498877

order	512
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#447
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1540 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)(7,19)(8,25)(9,15)(10,33)(11,17)(12,18)(13,22)(14,29)(16,36)(20,38)(21,52)(23,47)(24,53)(26,63)(27,62)(28,46)(30,48)(31,44)(32,50)(34,39)(35,43)(37,49)(40,59)(41,60)(42,56)(45,65)(51,55)(54,68)(57,67)(58,61)(64,66);(1,2)(3,4)(5,7)(6,13)(8,26)(9,21)(10,34)(11,23)(12,24)(14,42)(15,39)(16,47)(17,40)(18,41)(19,33)(20,30)(22,36)(25,61)(27,57)(28,54)(29,55)(31,56)(32,60)(35,64)(37,59)(38,43)(44,66)(45,67)(46,63)(48,68)(49,53)(50,52)(51,62)(58,65);(2,4)(5,8)(6,14)(7,20)(9,27)(10,35)(11,30)(12,31)(13,38)(15,43)(16,48)(17,45)(18,46)(19,51)(21,26)(22,58)(23,56)(24,57)(25,33)(28,32)(29,36)(34,55)(37,65)(39,63)(40,42)(41,67)(44,49)(47,61)(50,62)(52,66)(53,68)(54,59)(60,64);(5,9,12,11)(6,15,18,17)(7,21,24,23)(8,27,31,30)(10,32,37,36)(13,39,41,40)(14,43,46,45)(16,33,50,49)(19,52,53,47)(20,26,57,56)(22,34,60,59)(25,62,44,48)(28,65,29,35)(38,63,67,42)(51,66,68,61)(54,58,55,64)

Polycyclic presentation:

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

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, 406073);