#148218

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2054
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:

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

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,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=gh,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, 55414);