#494341

order	512
nilpotency class	4
isoclinism	isoclinic class of #483807
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1030 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)(5,6,11,17,12,19,37,18)(7,20,25,42,22,51,38,50)(8,26,29,60,30,61,39,46)(9,31,33,15,34,16,10,35)(13,24,43,23,41,32,44,36)(14,45,47,28,49,27,48,63)(21,53,57,65,54,67,58,40)(52,64,59,56,68,55,66,62);(2,4)(5,7,12,22)(6,13)(8,21,30,54)(9,32,34,24)(10,36,33,23)(11,38,37,25)(14,40)(15,50)(16,51)(17,44)(18,43)(19,41)(20,31)(26,59)(27,62,45,56)(28,64,63,55)(29,58,39,57)(35,42)(46,68)(47,67)(48,53)(49,65)(52,60)(61,66);(5,8,12,30)(6,14,19,49)(7,21,22,54)(9,28,34,63)(10,27,33,45)(11,39,37,29)(13,40,41,65)(15,26,35,61)(16,46,31,60)(17,48,18,47)(20,52,51,68)(23,56,36,62)(24,55,32,64)(25,58,38,57)(42,66,50,59)(43,67,44,53);(7,22)(13,41)(20,51)(21,54)(23,36)(24,32)(25,38)(40,65)(42,50)(43,44)(52,68)(53,67)(55,64)(56,62)(57,58)(59,66)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^b=di,e^b=eh,f^b=fhi,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, 401537);