Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1000 conjugacy classes of subgroups; only the 370 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 370 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,13)(9,25)(10,14)(11,15)(12,19)(17,20)(18,30)(21,27)(22,35)(23,28)(24,29)(26,32)(31,33)(34,36);(5,7,9,19,11,20,10,21)(6,12,14,16,15,27,25,17)(8,18,22,32,24,33,23,34)(13,26,28,30,29,36,35,31);(5,6)(7,17,20,16)(8,13)(9,25)(10,14)(11,15)(12,19,27,21)(18,31,33,30)(22,35)(23,28)(24,29)(26,32,36,34);(5,8,10,23,11,24,9,22)(6,13,14,28,15,29,25,35)(7,18,21,34,20,33,19,32)(12,26,16,30,27,36,17,31)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,f^b=fi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266182);