Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93165 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C23×C4 | C42 |
| Frattini, cosocle | C22×#123 | C22 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#123 ◁ G | |
| LECS | G ▷ C22×#123 ▷ C24 ▷ C22 ▷ C1 | |
548 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2,3,6)(4,7,8,5)(9,10)(11,19)(12,23,24,16)(13,17)(14,18)(15,20,27,33)(21,31)(22,32)(25,36,37,29)(26,30)(28,34,38,40)(35,39);(1,2)(3,6,8,5)(4,7)(9,11,13,21,14,22,26,35)(10,15,17,28,18,27,30,38)(12,20,25,34,24,33,37,40)(16,19,29,31,23,32,36,39)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,e^2=hi,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=gi,c^b=cg,d^b=dfi,f^b=fi,g^b=gi,d^c=dh,e^c=ei,e^d=ei44 conjugacy classes — view the full character table.
| maximal subgroup of | #72227719, #72227720, #72227724 |
| maximal normal subgroup of | #72227719, #72227720, #72227724 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364);