#148220

order	512
nilpotency class	4
isoclinism	isoclinic class of #13469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2065
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11076
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

671 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=ghi,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, 55416);