Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,20,33,19,24,18,23)(15,25,27,21,28,36,16,29)(17,22,34,43,32,39,31,38)(26,37,40,35,41,44,30,42);(2,4)(6,9)(10,11)(13,15)(14,21)(16,20)(17,30)(18,27)(19,28)(22,37)(23,36)(24,29)(25,33)(26,31)(32,40)(34,41)(35,38)(39,44)(42,43);(5,7,8,12)(6,10,9,11)(13,16,20,15,19,27,18,28)(14,21,33,36,24,29,23,25)(17,30,34,26,32,40,31,41)(22,35,43,44,39,42,38,37);(13,17,18,31,19,32,20,34)(14,22,23,38,24,39,33,43)(15,26,16,30,28,41,27,40)(21,35,25,37,29,42,36,44);(13,18,19,20)(14,23,24,33)(15,27,28,16)(17,31,32,34)(21,36,29,25)(22,38,39,43)(26,40,41,30)(35,44,42,37)Polycyclic presentation:
2^9|a^2=fg,c^2=gh,d^2=h,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373389);