Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476654 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1102 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,10,15,9,24)(7,12,20,36,19,30,25,29)(8,14,26,31,23,33,22,13)(17,34,35,27,21,32,18,28);(1,2)(3,4)(5,7,9,20,10,19,11,25)(6,12,16,29,15,30,24,36)(8,21,22,18,23,17,26,35)(13,28,14,32,31,27,33,34);(2,4)(5,8)(6,13,15,31)(7,17,19,21)(9,22)(10,23)(11,26)(12,27)(14,24,33,16)(18,20,35,25)(28,30)(29,34)(32,36);(5,8)(6,14,15,33)(7,18,19,35)(9,22)(10,23)(11,26)(12,28)(13,16,31,24)(17,25,21,20)(27,30)(29,32)(34,36)Polycyclic presentation:
2^9|a^2=efi,b^2=f,c^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,e^b=ei,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384370);