#147890

order	512
nilpotency class	4
isoclinism	isoclinic class of #129559
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

1110 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)(7,18)(8,23,25,40)(9,27)(10,33)(11,17)(12,20)(13,24,41,60)(14,31)(15,45)(16,30)(19,52,54,36)(21,46)(22,37)(26,44,35,47)(28,49)(29,42,62,50)(32,48)(34,43)(38,61)(39,57)(51,53,59,67)(55,65,58,66)(56,64,68,63);(1,2)(3,4)(5,7,11,22)(6,12,17,39)(8,19,25,54)(9,28,32,61)(10,34,30,21)(13,36,41,52)(14,43,15,46)(16,49,33,38)(18,48,37,27)(20,45,57,31)(23,59,40,51)(24,56,60,68)(26,58,35,55)(29,53,62,67)(42,66,50,65)(44,63,47,64);(2,4)(5,8)(6,13)(7,19)(9,29)(10,35)(11,25)(12,36)(14,44)(15,47)(16,50)(17,41)(18,51)(20,56)(21,58)(22,54)(23,48)(24,31)(26,30)(27,40)(28,53)(32,62)(33,42)(34,55)(37,59)(38,66)(39,52)(43,63)(45,60)(46,64)(49,65)(57,68)(61,67)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,c^b=cf,d^b=dh,e^b=eh,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, 55086);