Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,27,29,25)(20,23,30,32)(21,31,28,26);(1,2)(3,4)(9,11,12,15)(10,13,14,16)(17,19,28,30,22,29,21,20)(18,23,31,25,24,32,26,27);(2,4)(5,7,8,6)(10,14)(13,16)(18,24)(19,29)(20,30)(26,31);(17,20,21,29,22,30,28,19)(18,25,26,23,24,27,31,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=hi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281172);