Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,12)(7,17)(9,10)(13,28)(14,32)(15,26)(16,27)(18,25)(19,34)(20,23)(21,24)(29,36)(30,33)(31,35);(5,6,8,12)(7,18,19,35)(9,14,21,26)(10,15,24,32)(11,16,22,27)(13,29,30,23)(17,31,34,25)(20,33,36,28);(5,7,10,23,11,20,9,17)(6,13,15,33,16,31,14,18)(8,19,24,29,22,36,21,34)(12,25,32,35,27,30,26,28)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=ei,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56485);