Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
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 | |
2170 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)(7,10)(8,9)(11,12)(13,14,19,15,18,21,16,23)(17,25,27,28,26,20,24,22);(6,9)(7,11)(13,15,19,21,18,23,16,14)(17,22,27,25,26,28,24,20);(2,4)(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,15,21,23)(17,24,26,27)(20,22,25,28);(6,9)(10,12)(13,17)(14,20)(15,22)(16,24)(18,26)(19,27)(21,25)(23,28);(14,21)(15,23)(17,26)(24,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,e^b=ei,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282878);