Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,14,18,22,17,19,16,21)(15,20,25,28,24,26,23,27);(2,3)(5,7,9,8)(6,10,11,12)(14,19)(20,26)(21,22)(27,28);(5,8,9,7)(6,10,11,12)(13,15)(14,20)(16,23)(17,24)(18,25)(19,26)(21,27)(22,28);(1,3,4,2)(13,16,17,18)(14,21,19,22)(15,23,24,25)(20,27,26,28);(15,24)(20,26)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^b=dfi,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287809);