#416817

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15233
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

817 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=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, 324013);