#484222

order	512
nilpotency class	4
isoclinism	isoclinic class of #484217
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(5,8)(9,10)(11,23)(12,28)(13,19)(14,34)(15,21)(16,22)(17,26)(18,29)(20,36)(24,52)(25,50)(27,51)(30,61)(31,43)(32,45)(33,48)(35,40)(37,47)(38,46)(39,66)(41,65)(42,57)(44,63)(49,53)(54,58)(55,71)(56,70)(59,60)(62,68)(64,69)(67,72);(1,2)(3,5)(4,6)(7,8)(9,11,16,27)(10,17,22,42)(12,24,31,56)(13,25,15,38)(14,35,37,65)(18,39,45,68)(19,40,21,41)(20,46,48,50)(23,47,51,34)(26,33,57,36)(28,58,43,67)(29,53,32,55)(30,62,63,66)(44,70,61,52)(49,69,71,59)(54,64,72,60);(1,3)(2,5)(4,7)(6,8)(9,12)(10,18)(11,24)(13,32)(14,30)(15,29)(16,31)(17,39)(19,28)(20,44)(21,43)(22,45)(23,49)(25,55)(26,54)(27,56)(33,64)(34,59)(35,62)(36,60)(37,63)(38,53)(40,58)(41,67)(42,68)(46,70)(47,69)(48,61)(50,52)(51,71)(57,72)(65,66);(9,13,16,15)(10,19,22,21)(11,25,27,38)(12,29,31,32)(14,33,37,36)(17,40,42,41)(18,43,45,28)(20,34,48,47)(23,50,51,46)(24,53,56,55)(26,35,57,65)(30,60,63,64)(39,67,68,58)(44,69,61,59)(49,70,71,52)(54,66,72,62)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391418);