Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455107 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,28)(17,33)(18,22)(19,23)(21,30)(24,37)(25,42)(26,35)(27,36)(29,43)(31,38)(32,39)(34,40)(41,44);(6,9)(10,12)(13,15,18,26,19,27,17,25)(14,20,22,35,23,36,33,42)(16,29,31,41,32,40,30,24)(21,37,28,43,38,44,39,34);(5,7)(6,10)(8,11)(9,12)(13,16,19,32)(14,21,23,38)(15,24,27,41)(17,31,18,30)(20,34,36,43)(22,39,33,28)(25,29,26,40)(35,44,42,37);(2,4)(7,11)(10,12)Polycyclic presentation:
2^9|b^2=g,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 371225);