#149143

order	512
nilpotency class	4
isoclinism	isoclinic class of #13504
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2077
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11315
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

409 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fh,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi

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, 56339);