Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#3480 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3241 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)(6,9)(10,12)(13,14,19,24,18,22,17,23)(15,25,16,21,26,20,27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,27)(18,26)(21,23)(22,25)(24,28);(2,4)(13,16)(14,21)(15,17)(18,27)(19,26)(20,24)(22,28)(23,25);(6,9)(10,12)(14,22)(15,26)(16,27)(23,24);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,22,24)(15,16,26,27)(20,28,25,21)Polycyclic presentation:
2^9|a^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ci,d^b=df,e^b=ei,d^c=di,e^c=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286698);