#141447

order	512
nilpotency class	4
isoclinism	isoclinic class of #141445
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9783
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 325 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)(3,15,11,30)(4,20,24,52)(5,26,29,19)(6,31)(7,13)(8,14)(9,17)(10,23,37,58)(12,21)(16,43)(18,48,32,28)(22,55)(25,53,35,63)(27,60)(33,40)(34,36)(38,51,47,54)(39,64)(41,57)(42,44,49,50)(45,62)(46,61)(56,59);(1,3,15,2,9,19,29,13,8,11,30,14,7,26,5,17)(4,16,41,10,35,46,22,38,24,39,62,37,25,59,27,47)(6,32,42,40,21,44,48,36,34,18,49,12,33,50,28,31)(20,51,55,61,53,23,57,43,52,54,60,56,63,58,45,64);(1,4,21,54,7,25,6,23,8,24,33,51,9,35,34,58)(2,10,36,63,17,47,40,52,14,37,31,53,13,38,12,20)(3,16,44,60,26,59,32,57,11,39,50,55,19,46,18,45)(5,27,42,43,30,62,28,61,29,22,49,64,15,41,48,56)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=gh,e^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48643);