#147134

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#388
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)(3,14,17,9)(4,20)(5,25)(6,13)(7,12)(8,11)(10,22)(15,34,45,58)(16,32,19,42)(18,43,49,33)(21,41,47,31)(23,52)(24,37)(26,35)(27,55)(28,56)(29,39)(30,38)(36,54)(40,44)(46,62,50,63)(48,60,51,64)(53,61,57,59);(1,3,6,18,7,19,29,51,8,17,30,49,5,16,27,48)(2,9,13,33,12,32,39,60,11,14,38,43,25,42,55,64)(4,21,23,53,22,45,54,50,24,47,28,57,26,15,44,46)(10,34,36,62,37,41,56,61,35,58,40,63,20,31,52,59);(1,4)(2,10)(3,15,17,45)(5,26)(6,28)(7,22)(8,24)(9,31,14,41)(11,35)(12,37)(13,40)(16,47,19,21)(18,50,49,46)(20,25)(23,30)(27,54)(29,44)(32,34,42,58)(33,61,43,59)(36,38)(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,c^b=ch,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, 54330);