#146884

order	512
nilpotency class	4
isoclinism	isoclinic class of #128584
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10806
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

483 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,f^b=fi,e^c=ehi,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54080);