Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
1820 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)(13,14,19,21,18,20,16,25)(15,23,22,27,24,17,26,28);(5,6,9,10)(7,12,8,11)(13,15,18,24)(14,17,20,23)(16,26,19,22)(21,28,25,27);(2,4)(14,20)(15,24)(21,25)(22,26);(5,7,9,8)(6,11,10,12)(13,16,18,19)(14,21,20,25)(15,22,24,26)(17,27,23,28);(5,8,9,7)(6,11,10,12)(13,17,16,27,18,23,19,28)(14,22,25,15,20,26,21,24)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,c^b=ci,d^b=df116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6292362);