Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,9)(8,10)(11,12)(13,14,18,23,20,24,19,17)(15,25,28,21,27,16,22,26);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,21)(16,17)(18,22)(19,28)(20,27)(23,25)(24,26);(2,4)(6,10)(9,12)(13,16,19,21,20,25,18,26)(14,22,17,27,24,28,23,15);(2,4)(13,17,19,24,20,23,18,14)(15,26,22,16,27,21,28,25);(13,18,20,19)(14,23,24,17)(15,22,27,28)(16,21,25,26)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=f,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348952);