Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(13,14)(15,21)(16,20)(17,19)(18,22)(23,27)(24,26)(25,28);(2,4)(5,6,9,8)(7,11,12,10);(5,7,9,12)(6,10,8,11)(13,15)(14,18)(16,23)(17,22)(19,21)(20,25)(24,28)(26,27);(14,19)(18,21)(20,26)(25,27);(5,8,9,6)(7,10,12,11)(15,22)(18,21)(23,28)(25,27);(13,16,17,24)(14,20,19,26)(15,23,22,28)(18,25,21,27)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544486);