#148391

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11121
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

665 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)(3,15,20,48)(4,21,26,45)(5,28)(6,13)(7,33)(8,35)(9,14)(10,17,38,19)(11,24,42,27)(12,31)(16,50)(18,47,32,49)(22,61)(23,62,25,41)(29,43,34,60)(30,44)(36,51)(37,55,39,57)(40,53)(46,56)(52,59)(54,64)(58,63);(1,3,5,17,8,18,31,55,9,20,7,19,6,32,30,57)(2,10,12,15,13,39,28,49,14,38,44,48,35,37,33,47)(4,22,23,56,27,54,34,16,26,51,25,52,24,58,29,53)(11,40,41,64,21,59,60,61,42,50,62,63,45,46,43,36);(1,4)(2,11)(3,16)(5,29)(6,27)(7,34)(8,24)(9,26)(10,36)(12,43)(13,45)(14,42)(15,46)(17,54)(18,56)(19,58)(20,53)(21,35)(22,55)(23,30)(25,31)(28,62)(32,52)(33,41)(37,64)(38,61)(39,63)(40,47)(44,60)(48,59)(49,50)(51,57)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=dhi,e^c=ei,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, 55587);