Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,9)(7,11)(13,14,19,24,18,23,27,25)(15,17,26,16,21,20,22,28);(5,6)(7,10)(8,9)(11,12)(13,15,19,26,18,21,27,22)(14,20,24,28,23,17,25,16);(2,4)(6,9)(10,12)(13,16)(14,21)(15,23)(17,27)(18,28)(19,20)(22,24)(25,26);(5,7,8,11)(6,10,9,12)(13,17,18,20)(14,22,23,26)(15,25,21,24)(16,27,28,19);(15,21)(16,28)(17,20)(22,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,e^b=ei,e^c=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282910);