#494544

order	512
nilpotency class	4
isoclinism	isoclinic class of #33595
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1282 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)(5,6,12,19,37,41,36,17,13,20,11,18,9,16,10,38)(7,21,26,14,35,45,24,52,27,53,25,46,23,44,39,43)(8,28,32,50,29,48,30,15,34,49,31,47,33,54,40,51)(22,56,60,68,57,42,63,55,62,65,59,64,61,67,58,66);(5,7,9,35,13,27,37,23)(6,14,41,43,20,46,16,52)(8,22,29,61,34,62,33,57)(10,26,11,24,36,25,12,39)(15,42,50,65,51,67,47,56)(17,44,18,53,38,45,19,21)(28,64,54,66,49,68,48,55)(30,59,31,63,40,60,32,58);(2,4)(5,8)(6,15)(7,22)(9,29)(10,32)(11,30)(12,40)(13,34)(14,42)(16,47)(17,28)(18,54)(19,48)(20,51)(21,55)(23,57)(24,59)(25,63)(26,58)(27,62)(31,36)(33,37)(35,61)(38,49)(39,60)(41,50)(43,65)(44,64)(45,68)(46,67)(52,56)(53,66);(5,9,13,37)(6,16,20,41)(7,23,27,35)(8,29,34,33)(10,11,36,12)(14,43,46,52)(15,47,51,50)(17,19,38,18)(21,44,53,45)(22,57,62,61)(24,26,39,25)(28,48,49,54)(30,31,40,32)(42,65,67,56)(55,64,66,68)(58,60,63,59)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,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, 401740);