#420933

order	512
nilpotency class	3
isoclinism	isoclinic class of #420932
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15282
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1468 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,g^d=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328129);