Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14)(15,22)(16,21)(17,26)(18,27)(19,23)(20,25)(24,28);(2,3)(5,7)(6,10)(8,11)(9,12)(13,15,18,25,19,24,16,17)(14,20,21,22,23,26,27,28);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,16,19,18)(14,21,23,27)(15,17,24,25)(20,22,26,28);(1,3,4,2)(7,12)(10,11)(15,24)(17,25)(20,26)(22,28);(13,17,16,24,19,25,18,15)(14,22,27,20,23,28,21,26)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=h,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7511590);