#148611

order	512
nilpotency class	4
isoclinism	isoclinic class of #13497
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2075
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11199
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

631 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)(3,14,18,9)(4,19,23,54)(5,25)(6,13)(7,31)(8,11)(10,21,37,57)(12,28)(15,44)(16,47,50,59)(17,35,30,34)(20,55)(22,40,24,58)(26,38,56,41)(27,39)(29,53)(32,45)(33,49,43,61)(36,48)(42,60)(46,62)(51,52)(63,64);(1,3)(2,9)(4,20)(5,16)(6,30)(7,17)(8,18)(10,36)(11,14)(12,33)(13,35)(15,22)(19,52)(21,48)(23,51)(24,46)(25,59)(26,60)(27,61)(28,49)(29,50)(31,34)(32,38)(37,63)(39,43)(40,62)(41,42)(44,58)(45,56)(47,53)(54,55)(57,64);(1,4)(2,10,11,37)(3,15)(5,26)(6,24)(7,22)(8,23)(9,32,14,42)(12,40,39,58)(13,41,31,38)(16,48)(17,51)(18,46)(19,53,54,25)(20,30)(21,28)(27,57)(29,56)(33,55,43,52)(34,63,35,36)(44,59,62,47)(45,49)(50,64)(60,61)

Polycyclic presentation:

2^9|a^2=g,c^2=d,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,f^b=fi,g^b=gi,e^c=eh,f^c=fi,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, 55807);