#479785

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17246
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

494 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)(3,9,17,33)(4,19,22,41)(5,23,26,39)(6,13)(7,14)(8,12)(10,21,35,29)(11,27,38,25)(15,42)(16,36)(18,32,28,40)(20,37)(24,31)(30,46)(34,53)(43,56,47,64)(44,57)(45,58)(48,61)(49,59)(50,55)(51,62,54,60)(52,63);(1,3,7,28,8,17,6,18)(2,9,13,32,12,33,14,40)(4,15,21,45,22,44,29,46)(5,24,27,48,26,49,25,16)(10,30,41,57,35,58,19,42)(11,36,39,59,38,61,23,31)(20,51,52,43,53,54,55,47)(34,60,63,56,37,62,50,64);(1,4,6,21,8,22,7,29)(2,10,14,41,12,35,13,19)(3,15,28,46,17,44,18,45)(5,20,25,52,26,53,27,55)(9,30,32,42,33,58,40,57)(11,37,23,50,38,34,39,63)(16,47,24,54,48,43,49,51)(31,56,36,60,59,64,61,62);(1,5)(2,11,12,38)(3,16,17,48)(4,20)(6,25)(7,27)(8,26)(9,31)(10,34,35,37)(13,39,14,23)(15,43,44,47)(18,49,28,24)(19,50,41,63)(21,52)(22,53)(29,55)(30,56)(32,36)(33,59)(40,61)(42,60)(45,54,46,51)(57,62)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386981);