Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1441 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,11,14,9,23)(7,12,18,28,17,27,24,26)(8,19,22,30,20,35,21,13)(16,31,32,25,33,34,36,29);(1,2)(3,4)(5,7)(6,12,14,27)(8,16)(9,18)(10,24)(11,17)(13,29,30,25)(15,26,23,28)(19,34,35,31)(20,33)(21,32)(22,36);(2,4)(5,8,9,21,11,20,10,22)(6,13,15,19,14,30,23,35)(7,16,18,32,17,33,24,36)(12,25,26,31,27,29,28,34)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=gi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31102);