Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | #52164 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(7,11)(13,14,20,15,19,16,18,22)(17,21,26,23,27,24,28,25);(5,6)(7,10)(8,9)(11,12)(13,15,20,16,19,22,18,14)(17,25,26,21,27,23,28,24);(1,2,3,4)(13,16,20,22,19,14,18,15)(17,24,26,25,27,21,28,23);(6,9)(10,12)(13,17)(14,21)(15,23)(16,24)(18,28)(19,27)(20,26)(22,25);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,22,16,15)(17,26,27,28)(21,23,24,25)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=df,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303777);