Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| 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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(10,13)(14,16)(17,19,20,27,21,26,28,25)(18,22,24,30,23,31,32,29);(1,2,3,4)(11,15)(14,16)(19,25)(20,28)(22,29)(24,32)(26,27)(30,31);(9,11,12,15)(10,14,13,16)(18,23)(19,26)(24,32)(25,27)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,d^c=de,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288016);