Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #33175 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1947 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)(5,6,9,8)(7,10,12,11)(13,14,18,21)(15,22,25,28)(16,20,17,26)(19,23,27,24);(6,8)(10,11)(13,15,17,24,18,25,16,23)(14,19,20,22,21,27,26,28);(2,4)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,26)(17,20)(18,21)(22,24)(23,28)(25,27);(5,7,9,12)(6,10,8,11)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28);(5,8,9,6)(7,10,12,11)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6534540);