Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500829 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2206 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,16,22,32,41,33,40,19,25,17,23,18,24,31,21)(15,20,27,35,44,34,26,42,30,38,28,36,29,37,43,39);(6,9)(10,11)(13,15,16,27,32,44,33,26,19,30,17,28,18,29,31,43)(14,20,22,35,41,34,40,42,25,38,23,36,24,37,21,39);(2,4)(14,21)(15,26)(16,31)(17,33)(18,32)(20,34)(22,24)(23,41)(25,40)(27,44)(28,29)(30,43)(37,38)(39,42);(5,7,8,12)(6,10,9,11)Polycyclic presentation:
2^9|a^2=f,b^2=gi,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh,h^c=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413779);