Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457092 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(2,3)(5,6)(7,10)(8,11)(9,12)(13,14)(15,23)(16,30)(17,35)(18,24)(19,26)(20,25)(21,28)(22,33)(27,43)(29,38)(31,37)(32,41)(34,40)(36,42)(39,44);(1,2,4,3)(5,7,9,8)(6,10,12,11)(13,15,20,29,18,28,19,17)(14,21,25,35,24,23,26,38)(16,31,33,27,32,36,34,44)(22,39,41,37,40,43,30,42);(1,2)(3,4)(5,7,9,8)(6,11,12,10)(13,16,20,34,18,32,19,33)(14,22,26,41,24,40,25,30)(15,27,29,31,28,44,17,36)(21,37,38,43,23,42,35,39);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,17,19,28,18,29,20,15)(14,23,25,38,24,21,26,35)(16,31,33,27,32,36,34,44)(22,37,30,39,40,42,41,43)Polycyclic presentation:
2^9|b^2=fghi,c^2=ef,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380871);