Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509065 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(5,6)(7,8)(9,11)(10,12)(13,14,19,33,31,40,29,20,18,24,16,22,17,23,34,36)(15,25,42,37,32,44,30,21,28,43,26,41,27,39,35,38);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12);(2,4)(5,8,9,10)(6,7,11,12)(14,20)(16,29)(17,31)(19,34)(21,25)(22,23)(24,36)(26,30)(27,32)(33,40)(35,42)(37,44)(38,43)(39,41);(6,11)(7,12)(13,15,18,28)(14,21,24,38)(16,30,19,35)(17,32,31,27)(20,25,36,43)(22,39,33,44)(23,41,40,37)(26,34,42,29)Polycyclic presentation:
2^9|a^2=fi,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417066);