#149075

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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	#11293
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

445 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=hi,e^2=i,f^2=hi,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fhi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi

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, 56271);