Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476144 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
978 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 978 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:
(1,2,4,3)(5,6,11,16)(7,17,21,36)(8,22,24,33)(9,13,26,14)(10,27,28,15)(12,29,31,20)(18,30,23,32)(19,34,25,35);(1,2)(3,4)(5,7,10,20,11,21,28,29)(6,12,15,17,16,31,27,36)(8,23,9,25,24,18,26,19)(13,30,22,35,14,32,33,34);(1,3,4,2)(5,8,11,24)(6,13,16,14)(7,18,21,23)(9,10,26,28)(12,30,31,32)(15,33,27,22)(17,34,36,35)(19,20,25,29);(2,3)(5,9,10,24,11,26,28,8)(6,14,15,33,16,13,27,22)(7,19,20,23,21,25,29,18)(12,30,17,35,31,32,36,34)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=ef,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385394);