#469043

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16641
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

558 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=efg,c^2=g,d^2=ef,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,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376239);