Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
597 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,9,12)(7,16,17,29)(8,21,23,30)(10,24,18,15)(11,26,20,13)(14,31,28,19)(22,36,25,34)(27,33,35,32);(1,2,4,3)(6,12)(7,17)(11,20)(14,28)(15,24)(19,31)(27,35)(32,33);(1,3,4,2)(5,7,18,11,9,17,10,20)(6,13,24,29,12,26,15,16)(8,19,22,27,23,31,25,35)(14,30,32,36,28,21,33,34);(5,8)(6,14)(7,19)(9,23)(10,25)(11,27)(12,28)(13,30)(15,33)(16,34)(17,31)(18,22)(20,35)(21,26)(24,32)(29,36)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=eh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419389);