Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,17,21,32,38,41,30,19,23,33,29,18,22,28,16)(15,24,25,39,34,43,44,36,27,40,42,35,26,37,31,20);(5,7,9,8)(6,10,11,12)(13,15,19,27)(14,20,23,36)(16,24,30,40)(17,31,33,44)(18,34,32,26)(21,37,29,43)(22,39,38,35)(25,41,42,28);(5,8,9,7)(6,10,11,12);(13,16,28,22,18,29,33,23,19,30,41,38,32,21,17,14)(15,20,31,37,26,35,42,40,27,36,44,43,34,39,25,24)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=eh,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405245);