Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,25)(21,24)(22,26)(27,30)(28,31)(29,32);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,27)(18,23,24,30)(20,29,28,22)(25,32,31,26);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(18,24)(19,27)(20,28)(26,32);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,20,22,19,21,28,29,27)(18,25,26,23,24,31,32,30);(2,6)(7,8)Polycyclic presentation:
2^9|b^2=i,c^2=f,d^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7517409);