Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394275 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2184 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)(5,6,11,9)(7,12,18,16)(8,19,22,32)(10,24,27,14)(13,26,21,23)(15,33,20,36)(17,31,25,29)(28,35,30,34);(2,4)(5,7)(6,12)(8,20,22,15)(9,16)(10,25)(11,18)(13,30,21,28)(14,31)(17,27)(19,33,32,36)(23,34,26,35)(24,29);(5,8)(6,13)(7,15)(9,21)(10,26)(11,22)(12,28)(14,32)(16,30)(17,35)(18,20)(19,24)(23,27)(25,34)(29,36)(31,33);(5,9,11,6)(7,16,18,12)(8,21,22,13)(10,14,27,24)(15,30,20,28)(17,29,25,31)(19,26,32,23)(33,35,36,34)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312476);