#148163

order	512
nilpotency class	4
isoclinism	isoclinic class of #13417
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2041
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)(3,17)(4,23,28,47)(5,15)(6,14)(7,33)(8,12)(9,16)(10,19)(11,26,42,58)(13,32)(18,39,53,63)(20,46)(21,50)(22,48)(24,35,56,62)(25,41,29,45)(27,40,59,43)(30,44)(31,37)(34,38)(36,55)(49,52,60,57)(51,64,54,61);(1,3,5,19,9,22,8,34)(2,10,12,17,16,38,15,48)(4,24,25,57,28,56,29,52)(6,31,7,20,30,21,32,55)(11,39,40,61,42,63,43,64)(13,37,44,36,33,50,14,46)(18,26,51,27,53,58,54,59)(23,49,45,35,47,60,41,62);(1,4)(2,11)(3,18)(5,29)(6,27)(7,26)(8,25)(9,28)(10,35)(12,43)(13,45)(14,47)(15,40)(16,42)(17,49)(19,54)(20,56)(21,57)(22,53)(23,44)(24,55)(30,59)(31,52)(32,58)(33,41)(34,51)(36,39)(37,61)(38,62)(46,63)(48,60)(50,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,d^c=di,e^c=ehi,f^c=fh,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, 55359);