Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417601 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,14)(7,16,19,11)(8,21,23,32)(10,12,26,15)(13,25,31,24)(17,29,20,28)(18,34,22,36)(27,35,30,33);(1,2)(3,4)(5,7)(6,11)(8,22)(9,19)(10,17)(12,28)(13,30)(14,16)(15,29)(18,23)(20,26)(21,34)(24,33)(25,35)(27,31)(32,36);(6,12)(7,17)(10,26)(13,21)(14,15)(18,33)(19,20)(22,35)(24,25)(28,29)(31,32)(34,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,27)(12,21)(14,31)(15,32)(16,30)(17,33)(19,22)(20,35)(24,26)(28,36)(29,34)Polycyclic presentation:
2^9|a^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324809);