#145533

order	512
nilpotency class	4
isoclinism	isoclinic class of #143954
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1687
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10536
prime radical	#33716	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

331 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|c^2=d,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fh,g^b=gi,e^c=egh,f^c=fgh,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52729);