Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511887 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#459 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33520 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,15)(8,20)(9,24)(10,26)(11,16)(12,18)(13,17)(14,25)(19,33)(21,32)(22,36)(23,28)(27,29)(30,34)(31,35);(5,7,9,18,11,19,25,10)(6,12,14,15,16,26,24,33)(8,17,21,34,23,35,22,27)(13,28,29,32,31,20,30,36);(5,8)(6,13)(7,17)(9,21)(10,27)(11,23)(12,28)(14,29)(15,32)(16,31)(18,34)(19,35)(20,26)(22,25)(24,30)(33,36);(2,4)Polycyclic presentation:
2^9|b^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,h^a=hi,f^c=fh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250042, #72250058, #72250062 |
| maximal normal subgroup of | #72250042, #72250058, #72250062 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419083);