#472868

order	512
nilpotency class	3
isoclinism	isoclinic class of #472867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16839
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

792 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380064);