#467939

order	512
nilpotency class	3
isoclinism	isoclinic class of #467938
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16595
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

691 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)(4,21)(5,11,29,41)(6,13)(7,12)(8,14)(9,18)(10,24)(16,31)(17,47,50,43)(19,34)(20,33)(22,53)(23,39,55,52)(25,38)(26,37)(27,36,54,61)(28,42,30,40)(32,51,60,49)(35,45)(44,57,56,62)(46,58)(48,64,63,59);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,24,16,26,45,25,46)(5,17,30,51,29,50,28,49)(10,35,21,31,38,53,37,58)(11,32,42,43,41,60,40,47)(23,48,54,44,55,63,27,56)(36,59,52,57,61,64,39,62);(1,4,7,24,8,26,6,25)(2,10,13,21,14,38,12,37)(3,16,19,45,20,46,18,22)(5,27,28,23,29,54,30,55)(9,31,33,53,34,58,15,35)(11,39,40,36,41,52,42,61)(17,48,49,44,50,63,51,56)(32,59,47,57,60,64,43,62);(1,5,7,28,8,29,6,30)(2,11,13,40,14,41,12,42)(3,17,19,49,20,50,18,51)(4,23,24,54,26,55,25,27)(9,32,33,47,34,60,15,43)(10,36,21,52,38,61,37,39)(16,44,45,63,46,56,22,48)(31,57,53,64,58,62,35,59)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 375135);