Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×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×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
956 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,24)(21,23)(22,30)(26,29)(27,32)(28,31);(2,4)(9,11,13,10)(12,15,16,14)(18,23)(19,27)(20,28)(26,30);(1,2,3,4)(9,12,13,16)(10,14,11,15)(17,19,22,20,21,27,29,28)(18,24,26,25,23,31,30,32);(10,11)(14,15)(17,20,22,27,21,28,29,19)(18,25,26,31,23,32,30,24)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=efg,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293952);