#416753

order	512
nilpotency class	3
isoclinism	isoclinic class of #398772
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2439
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15230
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

889 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)(3,13,16,8)(4,9)(5,22)(6,25)(7,11)(10,23)(12,26)(14,27,18,34)(15,40,41,50)(17,29,43,36)(19,46)(20,31)(21,49)(24,51)(28,42,35,58)(30,48)(32,37)(33,52)(38,60,44,62)(39,57,45,54)(47,64)(53,63,56,61)(55,59);(1,3,6,17,7,16,26,43)(2,8,25,36,11,13,12,29)(4,18,21,45,20,14,37,39)(5,15,23,42,24,41,52,58)(9,27,49,57,31,34,32,54)(10,28,51,40,33,35,22,50)(19,44,48,63,47,38,59,61)(30,53,64,62,55,56,46,60);(1,4)(2,9)(3,14,16,18)(5,19)(6,21)(7,20)(8,27,13,34)(10,30)(11,31)(12,32)(15,38,41,44)(17,39,43,45)(22,46)(23,48)(24,47)(25,49)(26,37)(28,53,35,56)(29,54,36,57)(33,55)(40,62,50,60)(42,61,58,63)(51,64)(52,59);(1,5,6,23,7,24,26,52)(2,10,12,22,11,33,25,51)(3,15,17,42,16,41,43,58)(4,19,21,48,20,47,37,59)(8,28,29,50,13,35,36,40)(9,30,32,46,31,55,49,64)(14,38,39,61,18,44,45,63)(27,53,54,60,34,56,57,62)

Polycyclic presentation:

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

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, 323949);