Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1214 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,7,8)(9,10,12,11)(13,14,19,24,18,23,17,22)(15,25,26,21,16,20,27,28);(1,2)(3,4)(6,8)(13,15,17,27,18,16,19,26)(14,20,22,21,23,25,24,28);(2,4)(6,8)(13,16,18,15)(14,21,23,28)(17,27,19,26)(20,24,25,22);(6,8)(9,11,12,10)(13,17,18,19)(14,22,23,24)(15,26,16,27)(20,28,25,21)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=i,d^2=fh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285117);