#149132

order	512
nilpotency class	4
isoclinism	isoclinic class of #13502
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	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

421 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)(3,15)(4,22,27,43)(5,29)(6,11)(7,34)(8,14)(9,17)(10,25,44,26)(12,31)(13,30)(16,50,52,36)(18,47)(19,39)(20,37)(21,49)(23,56,54,48)(24,55,60,45)(28,61,59,42)(32,40)(33,46)(35,38)(41,57,63,51)(53,62,58,64);(1,3)(2,9,14,40)(4,23)(5,19)(6,32)(7,35)(8,21)(10,41,44,63)(11,37,46,38)(12,15,29,49)(13,47,34,39)(16,26)(17,33)(18,31)(20,30)(22,50,43,36)(24,58)(25,52)(27,54)(28,51)(42,62,61,64)(45,56,55,48)(53,60)(57,59);(1,4,5,24,30,59,33,26,8,27,31,60,7,28,6,25)(2,10,11,42,34,55,12,22,14,44,46,61,13,45,29,43)(3,16,17,51,35,58,19,54,21,52,32,57,20,53,18,23)(9,36,15,48,47,64,37,41,40,50,49,56,39,62,38,63)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,h^b=hi,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, 56328);