Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518920 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,19,15)(16,23,27,24)(17,25,21,18)(20,28,22,26);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,19,14)(16,24,27,23)(17,18,21,25)(20,26,22,28);(6,9)(10,11)(13,16,25,28,19,27,18,26)(14,20,17,24,15,22,21,23);(5,7,8,12)(6,10,9,11)(13,17,18,14,19,21,25,15)(16,24,26,20,27,23,28,22);(1,3,4,2)(13,18,19,25)(14,17,15,21)(16,26,27,28)(20,24,22,23)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=hi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504004);