#416473

order	512
nilpotency class	3
isoclinism	isoclinic class of #398772
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2439
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15225
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

693 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323669);