#148536

order	512
nilpotency class	4
isoclinism	isoclinic class of #13490
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2073
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11189
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

647 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)(3,17,22,47)(4,24)(5,14)(6,12)(7,34)(8,35)(9,16)(10,19,40,31)(11,27)(13,30)(15,33)(18,50,52,37)(20,41,21,48)(23,38,36,39)(25,49,54,64)(26,55)(28,43)(29,46)(32,45)(42,51,62,60)(44,58)(53,61,57,63)(56,59);(1,3)(2,10)(4,25)(5,21)(6,31)(7,19)(8,36)(9,22)(11,42)(12,38)(13,17)(14,47)(15,48)(16,40)(18,27)(20,30)(23,33)(24,37)(26,57)(28,60)(29,54)(32,52)(34,39)(35,41)(43,61)(44,49)(45,62)(46,50)(51,59)(53,58)(55,64)(56,63);(1,4,5,26,8,28,6,32,9,29,30,58,33,59,7,27)(2,11,12,43,15,44,14,24,16,45,34,56,35,55,13,46)(3,18,19,51,23,53,20,54,22,52,31,60,36,57,21,25)(10,37,17,49,41,61,39,42,40,50,47,64,48,63,38,62)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=hi,f^2=hi,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=fh,g^b=gi,h^b=hi,e^c=ei

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, 55732);