#477839

order	512
nilpotency class	3
isoclinism	isoclinic class of #477838
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

671 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385035);