Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| 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 | |
1122 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)(11,15)(12,13)(14,16)(17,18,20,24,21,23,29,31)(19,25,27,30,28,32,26,22);(1,2)(3,4)(17,19,21,28)(18,22,23,30)(20,26,29,27)(24,32,31,25);(2,4)(5,7,8,6)(10,13)(14,15)(18,23)(22,30)(24,31)(25,32);(9,11,12,16)(10,14,13,15)(18,24)(19,26)(20,29)(22,30)(23,31)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ehi,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296868);