#149263

order	512
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11328
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,d^c=dh,f^c=fh,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, 56459);