#475232

order	512
nilpotency class	3
isoclinism	isoclinic class of #469927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2605
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17035
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

533 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382428);