#148646

order	512
nilpotency class	4
isoclinism	isoclinic class of #136630
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11200
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

623 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=dh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,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, 55842);