Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C43 |
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×#376 ◁ 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:
(1,2)(3,4)(5,6,8,7)(9,10)(11,14)(12,16)(13,15)(18,22)(19,25)(20,27)(23,30)(24,29)(26,31);(5,7,8,6)(9,11,13,12)(10,14,15,16)(17,18,20,22,21,24,27,29)(19,23,25,28,26,30,31,32);(9,12,13,11)(10,14,15,16)(19,26)(23,30)(25,31)(28,32);(2,4)(17,19,20,25,21,26,27,31)(18,23,22,28,24,30,29,32)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300604);