#147895

order	512
nilpotency class	4
isoclinism	isoclinic class of #129565
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,g^b=gi,h^b=hi,d^c=dh,e^c=eh,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, 55091);