Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | #35273 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2292 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)(5,6)(7,8)(9,11)(10,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,28)(25,26);(5,7)(6,10)(8,11)(9,12)(13,14,16,19)(15,22,23,28)(17,26,25,20)(18,27,21,24);(2,4)(5,8,9,10)(6,7,11,12)(13,15)(14,18)(16,23)(17,27)(19,21)(20,28)(22,26)(24,25);(2,4)(6,11)(7,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,24)(25,26)(27,28);(14,19)(18,21)(20,26)(22,28)Polycyclic presentation:
2^9|b^2=i,c^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408425);