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 | |
570 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)(14,19)(15,23)(16,28)(17,31)(20,24)(21,40)(22,36)(26,27)(29,33)(30,42)(32,34)(35,41)(38,44)(39,43);(5,7,9,8)(6,10,11,12)(13,14,18,22)(15,24,27,43)(16,29,32,40)(17,33,31,21)(19,34,36,28)(20,37,39,25)(23,35,26,44)(30,41,42,38);(5,8,9,7)(6,10,11,12);(13,15,18,27)(14,20,22,39)(16,30,32,42)(17,23,31,26)(19,35,36,44)(21,41,33,38)(24,40,43,29)(25,34,37,28)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405198);