#147136

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#387
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10875
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54332);