#416472

order	512
nilpotency class	3
isoclinism	isoclinic class of #416472
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2864
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15225
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

687 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:

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

Polycyclic presentation:

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

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, 323668);