Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,18,20,24,21,25,28,22)(19,23,26,30,27,31,32,29);(1,2)(3,4)(9,11,12,15)(10,13,14,16)(18,22)(20,28)(23,29)(24,25)(26,32)(30,31);(2,4)(10,14)(13,16)(17,19,21,27)(18,23,25,31)(20,26,28,32)(22,29,24,30);(5,7,8,6)(18,24)(20,28)(22,25)(23,30)(26,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=ehi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 283675);