#147919

order	512
nilpotency class	4
isoclinism	isoclinic class of #136733
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
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

948 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=gi,h^b=hi,d^c=dh,f^c=fh,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, 55115);