#149272

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
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	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

499 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=e,d^2=hi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,f^b=fi,d^c=dhi,f^c=fhi,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, 56468);