#147972

order	512
nilpotency class	3
isoclinism	isoclinic class of #2032
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#393
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11042
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

667 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55168);