Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #147859 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10995 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #60753 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11)(3,12,16,28)(4,17)(5,8,21,10)(7,22)(9,20)(13,30,15,29)(14,24)(18,25)(19,26)(23,32)(27,31);(1,3)(2,7)(4,14)(5,13)(6,16)(8,23)(9,12)(10,27)(11,24)(15,21)(17,28)(18,32)(19,31)(20,22)(25,30)(26,29);(2,8)(3,13)(5,21)(9,26)(10,11)(12,28)(14,31)(15,16)(17,25)(18,19)(22,32)(23,27)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55055);