Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190028 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1830 conjugacy classes of subgroups; only the 274 normal subgroups are shown.
This group has 1830 conjugacy classes of subgroups — too many to draw. Only its 274 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 — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,24)(13,18)(14,29)(15,19)(17,27)(20,33)(21,31)(22,32)(23,37)(25,30)(26,38)(28,34)(35,40)(36,39);(2,5)(6,8)(9,11)(10,16)(12,25,27,20)(13,21)(14,23)(15,22)(17,33,24,30)(18,31)(19,32)(26,36,28,35)(29,37)(34,40,38,39);(1,3)(2,6)(4,7)(5,8)(9,12,14,28,15,27,13,26)(10,17,18,34,19,24,29,38)(11,20,23,36,22,25,21,35)(16,30,31,39,32,33,37,40);(3,7)(6,8)(9,13,15,14)(10,18,19,29)(11,21,22,23)(12,26,27,28)(16,31,32,37)(17,34,24,38)(20,35,25,36)(30,39,33,40)Polycyclic presentation:
2^9|b^2=f,c^2=h,d^2=i,h^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284804, #72284849, #72284899, … |
| maximal normal subgroup of | #72284804, #72284849, #72284899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289906);