Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455828 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,15,10,17,12,16)(7,8,19,23,21,13,20,24)(9,25,28,14,27,30,29,31)(18,32,33,26,35,36,34,22);(1,2)(3,4)(5,7,11,20,10,21,12,19)(6,13,16,24,17,8,15,23)(9,18,28,34,27,35,29,33)(14,26,25,36,31,22,30,32);(2,4)(5,8,10,13)(6,7,17,21)(9,26,27,22)(11,24,12,23)(14,18,31,35)(15,20,16,19)(25,33,30,34)(28,32,29,36);(5,9)(6,14)(7,18)(8,22)(10,27)(11,28)(12,29)(13,26)(15,30)(16,25)(17,31)(19,33)(20,34)(21,35)(23,32)(24,36)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363863);