Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1874 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,11,17,9,15)(7,18,21,13,22,33,20,34)(8,23,26,29,27,35,25,14)(12,28,31,36,32,24,30,19);(5,7)(6,12,17,32)(8,24)(9,20)(10,21)(11,22)(13,29,34,14)(15,30,16,31)(18,23,33,35)(19,26)(25,36)(27,28);(1,2,3,4)(5,7)(6,13)(8,19)(9,21)(10,20)(11,22)(12,29)(14,32)(15,33)(16,18)(17,34)(23,31)(24,26)(25,28)(27,36)(30,35)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dhi,h^b=hi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249880, #72249881, #72249899, … |
| maximal normal subgroup of | #72249880, #72249881, #72249899, … |
| maximal quotient of | #72265700, #72279957, C2×#147475 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54671);