Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,26,23,27,30,31)(19,28,20,25,29,32,21,24);(1,2)(3,4)(9,11)(10,13)(12,16)(14,15)(17,19,23,29)(18,24,27,25)(20,22,21,30)(26,32,31,28);(2,4)(10,14)(13,15)(17,20,22,29,23,21,30,19)(18,25,26,32,27,24,31,28);(6,8)(11,16)(13,15)(17,21,22,19,23,20,30,29)(18,25,26,32,27,24,31,28)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=ef,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 338660);