#148148

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2067
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

679 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)(3,14)(4,20,24,40)(5,25)(6,29)(7,13)(8,16)(9,23,41,61)(10,28)(11,27)(12,46)(15,50,52,64)(17,34)(18,37)(19,36)(21,57,58,49)(22,45,26,48)(30,47)(31,42,60,39)(32,43)(33,59,63,55)(35,56)(38,53,62,51)(44,54);(1,3,5,16,7,19,28,18)(2,8,10,14,13,37,25,36)(4,21,22,55,24,58,26,59)(6,30,27,17,32,54,46,56)(9,38,39,50,41,62,42,64)(11,44,29,34,12,47,43,35)(15,31,51,23,52,60,53,61)(20,33,48,57,40,63,45,49);(1,4)(2,9)(3,15)(5,26)(6,31)(7,24)(8,33)(10,42)(11,45)(12,48)(13,41)(14,49)(16,53)(17,55)(18,51)(19,52)(20,43)(21,54)(22,28)(23,46)(25,39)(27,61)(29,40)(30,58)(32,60)(34,64)(35,50)(36,57)(37,63)(38,44)(47,62)(56,59)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,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=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, 55344);