#144660

order	512
nilpotency class	4
isoclinism	isoclinic class of #144660
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10393
commutator, Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 282 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51856);