Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30056 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2631 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1222 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)(5,6,9,14,10,15,23,12)(7,16,18,26,20,24,19,11)(8,13,21,28,22,29,34,27)(17,30,31,36,33,35,32,25);(5,7)(6,11)(8,17)(9,19)(10,20)(12,16)(13,25)(14,24)(15,26)(18,23)(21,32)(22,33)(27,30)(28,35)(29,36)(31,34);(2,4)(6,12)(8,21)(9,23)(11,24)(14,15)(16,26)(17,31)(18,19)(22,34)(27,28)(30,35)(32,33);(5,8,9,21,10,22,23,34)(6,13,14,28,15,29,12,27)(7,17,19,32,20,33,18,31)(11,25,24,35,26,36,16,30)Polycyclic presentation:
2^9|a^2=h,d^2=efhi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284999, #72285452, #72285600 |
| maximal normal subgroup of | #72284999, #72285452, #72285600 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289087);