Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(6,10)(11,12)(13,14,20,22,19,24,17,28)(15,25,16,26,23,21,27,18);(5,6,9,10)(7,12,8,11)(13,15,19,23)(14,21,24,25)(16,17,27,20)(18,28,26,22);(2,4)(13,16)(14,18)(15,17)(19,27)(20,23)(21,28)(22,25)(24,26);(5,7,9,8)(6,11,10,12)(13,17,19,20)(14,22,24,28)(15,16,23,27)(18,21,26,25);(5,8,9,7)(6,11,10,12)(13,18,17,21,19,26,20,25)(14,23,28,16,24,15,22,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,d^b=df,e^b=ei,d^c=di,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6292279);