#499184

order	512
nilpotency class	4
isoclinism	isoclinic class of #498860
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1004 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)(7,20)(8,26)(9,16)(10,15)(11,36)(12,19)(13,22)(14,30)(17,35)(18,33)(21,38)(23,50)(24,48)(25,40)(27,59)(28,58)(29,46)(31,64)(32,45)(34,47)(37,39)(41,56)(42,55)(43,53)(44,62)(49,63)(51,65)(52,66)(54,57)(60,67)(61,68);(5,7,12,25)(6,13,19,42)(8,27,28,52)(9,23,34,56)(10,22,35,55)(11,37,33,24)(14,43,44,68)(15,39,17,48)(16,20,47,40)(18,50,36,41)(21,29,51,31)(26,57,58,67)(30,54,62,60)(32,53,63,61)(38,49,65,45)(46,59,64,66);(2,4)(5,8,9,29,33,63,35,62,12,28,34,31,11,32,10,30)(6,14,15,45,36,64,47,58,19,44,17,49,18,46,16,26)(7,21,22,52,37,60,56,61,25,51,55,27,24,54,23,53)(13,38,20,43,50,57,48,59,42,65,40,68,41,67,39,66);(8,28)(14,44)(21,51)(26,58)(27,52)(29,31)(30,62)(32,63)(38,65)(43,68)(45,49)(46,64)(53,61)(54,60)(57,67)(59,66)

Polycyclic presentation:

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

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, 406380);