#479792

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17251
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

516 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)(3,15)(4,20)(5,11,28,40)(6,13)(7,12)(8,14)(9,18)(10,22)(16,45)(17,44,24,55)(19,42)(21,53,54,37)(23,41)(25,58,59,34)(26,39,27,38)(29,35)(30,33)(31,49)(32,57,36,51)(43,48)(46,60,64,63)(47,52)(50,62,56,61);(1,3)(2,9,14,33)(4,16,23,49)(5,24,28,17)(6,18)(7,30)(8,19)(10,31)(11,36)(12,15,13,42)(20,52)(21,46)(22,47,29,48)(25,50)(26,57,27,51)(32,40)(34,60,58,63)(35,45)(37,61,53,62)(38,55)(39,44)(41,43)(54,64)(56,59);(1,4,7,22,8,23,6,29)(2,10,13,20,14,35,12,41)(3,16,18,48,19,49,30,47)(5,25,27,54,28,59,26,21)(9,31,15,43,33,45,42,52)(11,37,39,58,40,53,38,34)(17,50,51,46,24,56,57,64)(32,61,44,60,36,62,55,63);(1,5,6,26,8,28,7,27)(2,11,12,38,14,40,13,39)(3,17,18,51,19,24,30,57)(4,21,22,25,23,54,29,59)(9,32,15,44,33,36,42,55)(10,34,20,37,35,58,41,53)(16,46,47,50,49,64,48,56)(31,60,52,61,45,63,43,62)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386988);