Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| 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 | |
1570 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,30)(21,32)(22,24)(23,29)(25,31)(26,28);(9,11)(10,13)(12,16)(14,15)(17,19,21,29,22,20,28,31)(18,23,26,27,24,25,32,30);(1,2,3,4)(11,16)(13,15)(18,24)(23,25)(26,32)(27,30);(10,14)(13,15)(17,20,21,31,22,19,28,29)(18,25,26,30,24,23,32,27)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=fi,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271526);