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 | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,28)(19,26,31,24)(20,32,30,25)(21,29,23,27);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,30)(22,31)(25,27)(26,28)(29,32);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,19,21,20,22,31,23,30)(18,25,27,24,28,32,29,26);(9,11,12,16)(10,14,13,15)(17,20,23,19,22,30,21,31)(18,26,29,32,28,24,27,25);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,27,28,29)(19,30,31,20)(24,25,26,32)Polycyclic presentation:
2^9|a^2=f,c^2=fhi,d^2=g,e^2=f,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^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504997);