Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,4)(9,10)(11,12)(13,15)(14,16)(17,18,22,23,21,25,19,26)(20,28,27,32,30,31,29,24);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,28,32,31);(2,4)(6,8)(9,12,13,14)(10,11,15,16)(18,23)(19,22)(24,31)(25,26)(27,29)(28,32);(2,4)(10,15)(11,16)(17,20,21,30)(18,24,25,32)(19,27,22,29)(23,31,26,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ei,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290489);