#146152

order	512
nilpotency class	4
isoclinism	isoclinic class of #143954
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1687
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10650
prime radical	C₂³	#473
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

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

normal subgroups only

This group has 185 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=dg,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fhi,g^c=gi,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, 53348);