Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476293 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,9,13)(7,17,18,26,21,28,20,12)(8,22,24,14,25,30,23,31)(19,29,33,36,35,27,34,32);(1,2)(3,4)(5,7,9,20,11,21,10,18)(6,12,15,17,16,26,13,28)(8,19,23,34,25,35,24,33)(14,29,30,36,31,27,22,32);(2,4)(6,13)(7,18)(8,23)(9,10)(12,26)(15,16)(20,21)(22,30)(24,25)(27,36)(29,32)(33,34);(5,8,10,24,11,25,9,23)(6,14,13,22,16,31,15,30)(7,19,18,33,21,35,20,34)(12,27,28,36,26,29,17,32)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=ei,e^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,d^b=def,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383898);