#470937

order	512
nilpotency class	3
isoclinism	isoclinic class of #470936
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16718
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

664 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^d=fi,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, 378133);