Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| 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×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2211 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,19,15,18,21,28,23)(16,24,26,25,17,20,27,22);(6,9)(10,12)(13,15,19,21,18,23,28,14)(16,25,26,20,17,22,27,24);(1,2,3,4)(13,16,18,17)(14,20,21,24)(15,22,23,25)(19,26,28,27);(5,7,8,11)(6,10,9,12)(13,17,18,16)(14,20,21,24)(15,22,23,25)(19,27,28,26);(16,17)(20,24)(22,25)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ci,e^c=ei,e^d=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279360);