#498879

order	512
nilpotency class	4
isoclinism	isoclinic class of #498812
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	#60643	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1530 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:

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

Polycyclic presentation:

2^9|a^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,d^b=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406075);