#144658

order	512
nilpotency class	4
isoclinism	isoclinic class of #144658
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10392
commutator, Frattini, cosocle	C₂²×#41	C₂³
prime radical	#3485	C₂²
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

412 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=dg,b^2=d,d^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,h^b=hi,d^c=dg,h^c=hi,e^d=ei,f^e=fi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51854);