Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #189517 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1436 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,21,25,13)(9,12)(10,27)(11,16)(14,20)(15,18)(19,31,22,33)(23,30,36,28)(24,32,34,35)(26,29);(1,2,3,4)(5,7,18,26,11,20,10,9)(6,12,27,14,16,29,15,17)(8,22,34,23,25,19,24,36)(13,30,35,33,21,28,32,31);(2,4)(5,8)(6,13)(7,19)(9,23)(10,24)(11,25)(12,28)(14,31)(15,32)(16,21)(17,33)(18,34)(20,22)(26,36)(27,35)(29,30);(5,9,10,20,11,26,18,7)(6,14,15,12,16,17,27,29)(8,23,24,22,25,36,34,19)(13,31,32,28,21,33,35,30)Polycyclic presentation:
2^9|a^2=h,b^2=eg,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269231);