Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
2025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14,19,16,18,21,27,23)(15,17,20,25,24,26,28,22);(5,7,10,6)(8,11,9,12)(13,15,19,20,18,24,27,28)(14,17,16,25,21,26,23,22);(2,4)(13,16,19,21,18,23,27,14)(15,22,20,17,24,25,28,26);(5,8,10,9)(6,11,7,12)(13,17)(14,20)(15,23)(16,24)(18,26)(19,25)(21,28)(22,27);(5,9,10,8)(6,11,7,12)(14,21)(16,23)(17,26)(22,25)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=dfi,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6303014);