#417105

order	512
nilpotency class	3
isoclinism	isoclinic class of #416736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2862
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15242
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324301);