Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476175 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23406 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1570 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 1570 conjugacy classes of subgroups — too many to draw. Only its 186 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:
(5,6,10,15,11,16,9,12)(7,17,20,29,21,27,19,13)(8,22,25,32,26,30,24,14)(18,34,35,33,36,28,23,31);(1,2)(3,4)(6,12,16,15)(13,27,29,17)(14,30,32,22)(28,33,34,31);(2,4)(5,7)(6,13)(8,23)(9,20)(10,19)(11,21)(12,17)(14,31)(15,27)(16,29)(18,24)(22,28)(25,36)(26,35)(30,34)(32,33);(5,8,11,26)(6,14,16,32)(7,18,21,36)(9,25,10,24)(12,22,15,30)(13,28,29,34)(17,33,27,31)(19,35,20,23)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384220);