#146829

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	#10796
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

447 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gi,c^b=ch,d^b=di,f^c=fi,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, 54025);