Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2482 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1793 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)(5,6,9,11,10,15,24,14)(7,16,17,25,19,12,23,29)(8,20,33,26,22,13,21,28)(18,30,34,36,32,27,31,35);(2,4)(6,11,15,14)(7,17,19,23)(12,16)(13,26,20,28)(18,31,32,34)(25,29)(27,30)(35,36);(5,7)(6,12)(8,18)(9,23)(10,19)(11,25)(13,27)(14,29)(15,16)(17,24)(20,30)(21,34)(22,32)(26,35)(28,36)(31,33);(5,8)(6,13)(7,18)(9,21)(10,22)(11,26)(12,27)(14,28)(15,20)(16,30)(17,31)(19,32)(23,34)(24,33)(25,35)(29,36)Polycyclic presentation:
2^9|a^2=ef,b^2=h,e^2=i,g^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,e^c=ei,g^c=gi,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250213, #72257147, #72257167 |
| maximal normal subgroup of | #72250213, #72257147, #72257167 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312963);