Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D8 |
| commutator | C8 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,7,8)(14,19)(15,23)(16,25)(17,24)(20,26)(21,22)(27,28);(2,4)(13,14,16,21,17,22,15,19,18,20,23,28,24,27,25,26);(9,10,11,12)(14,20)(19,26)(21,28)(22,27)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=h,h^2=i,b^a=bd,d^a=dh,f^a=fhi,h^a=hi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72322446, #72322452, #72322458, … |
| maximal normal subgroup of | #72322446, #72322452, #72322458, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53524);