#148123

order	512
nilpotency class	4
isoclinism	isoclinic class of #13374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2023
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11066
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

677 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fhi,g^c=gi,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, 55319);