Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,22,25,30)(20,27,28,24)(26,32,31,29);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,25)(24,27)(26,31);(11,15)(13,16)(17,19,22,23,21,25,30,18)(20,26,29,27,28,31,32,24);(10,14)(13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,27)(25,31)(30,32);(20,28)(24,27)(26,31)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341031);