Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,12)(9,10)(13,14,19,24)(15,22,26,21)(16,27,25,28)(17,20,23,18);(1,2)(3,4)(6,10)(11,12)(13,15,19,26)(14,21,24,22)(16,18,25,20)(17,27,23,28);(5,7,9,8)(6,11,10,12)(13,16,20,15,19,25,18,26)(14,22,17,27,24,21,23,28);(2,4)(5,7,9,8)(6,11,10,12)(13,17,20,24,19,23,18,14)(15,22,25,27,26,21,16,28);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,23,24,17)(15,25,26,16)(21,28,22,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=fgi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7472700);