#148111

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
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

681 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=eh,f^c=fhi,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, 55307);