Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,7,8)(10,13)(11,15)(18,22)(19,25)(20,28)(23,29)(24,30)(27,31);(2,4)(9,10,12,13)(11,14,15,16)(17,18)(19,26)(20,29)(21,23)(22,28)(24,25)(27,32)(30,31);(1,2,3,4)(9,11,12,15)(10,14,13,16)(18,23)(22,29)(24,30)(26,32);(10,13)(14,16)(17,19,20,25,21,27,28,31)(18,24,22,26,23,30,29,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fgi,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294149);