Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,12,8,10)(13,14,17,21)(15,22,24,27)(16,20,18,25)(19,26,28,23);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14)(15,19)(16,25)(17,21)(18,20)(22,23)(24,28)(26,27);(2,4)(5,8,9,7)(6,10,11,12)(13,15,17,24)(14,19,21,28)(16,23,18,26)(20,27,25,22);(2,4);(13,16,17,18)(14,20,21,25)(15,23,24,26)(19,27,28,22)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346384);