Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| 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 | |
536 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)(13,14,18,23,34,42,31,16,20,25,17,22,19,24,36,32)(15,26,28,39,37,43,33,21,30,38,27,44,29,41,35,40);(5,7,9,8)(6,10,11,12)(13,15,20,30)(14,21,25,40)(16,26,32,38)(17,33,18,35)(19,37,34,29)(22,41,23,43)(24,44,42,39)(27,36,28,31);(5,8,9,7)(6,10,11,12);(13,16,31,24,19,23,18,25,20,32,36,42,34,22,17,14)(15,21,33,41,29,39,28,38,30,40,35,43,37,44,27,26)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^a=di,d^b=de,e^b=eh,g^b=gh,h^b=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405251);