Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394048 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,11)(7,12,8,10)(13,14,19,20,18,22,15,23)(16,25,24,21,17,27,26,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,19)(14,20,22,23)(16,26,17,24)(21,27,28,25);(2,4)(13,16,18,17)(14,21,22,28)(15,24,19,26)(20,25,23,27);(5,8,9,7)(6,10,11,12)(13,17,18,16)(14,21,22,28)(15,24,19,26)(20,27,23,25)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^c=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303645);