Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432833 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1748 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)(11,15)(14,16)(17,18,21,23)(19,27,28,32)(20,26,22,25)(24,30,31,29);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(18,23)(19,28)(25,26)(29,30);(10,13)(14,16)(17,19)(18,24)(20,29)(21,28)(22,30)(23,31)(25,27)(26,32);(9,11,12,15)(10,14,13,16)(18,23)(24,31)(25,26)(27,32);(17,20,21,22)(18,25,23,26)(19,29,28,30)(24,27,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342393);