Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
789 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14)(7,16)(8,20)(10,13,23,15)(11,19)(12,21)(17,27,30,24)(18,26)(22,28)(25,32)(29,33)(31,34,36,35);(2,4)(5,7,10,18,9,19,23,32)(6,11,13,25,14,16,15,26)(8,17,22,31,21,30,33,36)(12,27,29,34,20,24,28,35);(1,2,3,4)(5,8)(6,12,14,20)(7,17)(9,21)(10,22)(11,24,16,27)(13,29,15,28)(18,31)(19,30)(23,33)(25,35,26,34)(32,36);(6,13,14,15)(8,21)(11,25,16,26)(12,28,20,29)(17,30)(22,33)(24,34,27,35)(31,36)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=g,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72252691, #72256865, #72256881, … |
| maximal normal subgroup of | #72252691, #72256865, #72256881, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 317608);