Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #137768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×2−1+4 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1307 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6)(7,11)(8,19,21,12)(9,13)(10,14)(15,28)(16,32,34,24)(17,25)(18,26)(20,29,35,27)(22,30)(23,31,36,33);(1,2)(3,4)(5,7,10,18)(6,11,14,26)(8,16,21,34)(9,15,22,17)(12,24,19,32)(13,28,30,25)(20,31,35,33)(23,29,36,27);(1,3,4,2)(6,12)(7,15)(9,22)(11,23)(13,29)(14,19)(16,31)(17,18)(20,35)(24,28)(25,32)(26,36)(27,30)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=d,f^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,f^b=fi,f^c=fi,g^c=gh,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55777);