#148305

order	512
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#11095
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

645 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=di,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, 55501);