#476442

order	512
nilpotency class	3
isoclinism	isoclinic class of #476441
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17089
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

804 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383638);