Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| 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 | |
1032 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)(11,15)(12,13)(14,16)(17,18)(19,26)(20,27)(21,31)(22,25)(23,28)(24,29)(30,32);(2,4)(5,7,8,6)(10,13)(11,16)(17,19,22,29,23,30,20,21)(18,24,27,26,28,31,25,32);(2,4)(9,11,12,16)(10,14,13,15)(17,20,23,22)(18,25,28,27)(19,29,30,21)(24,26,31,32);(2,4)(10,13)(14,15)(17,21,20,30,23,29,22,19)(18,26,25,24,28,32,27,31)Polycyclic presentation:
2^9|a^2=h,b^2=eghi,c^2=g,d^2=ef,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, 278001);