Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2301 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)(5,6,8,9)(7,10,11,12)(13,14)(15,26)(16,22)(17,23)(18,37)(19,25)(20,24)(21,30)(27,28)(29,41)(31,44)(32,35)(33,36)(34,38)(39,40)(42,43);(6,9)(10,12)(13,15,19,30)(14,21,25,26)(16,32,23,43)(17,35,22,42)(18,31,20,29)(24,44,37,41)(27,33,40,34)(28,38,39,36);(2,4)(13,16,19,23)(14,22,25,17)(15,27,30,40)(18,34,20,33)(21,39,26,28)(24,36,37,38)(29,32,31,43)(35,44,42,41);(5,7,8,11)(6,10,9,12)(13,17,19,22)(14,23,25,16)(15,28,30,39)(18,36,20,38)(21,40,26,27)(24,34,37,33)(29,35,31,42)(32,44,43,41);(13,18,19,20)(14,24,25,37)(15,29,30,31)(16,33,23,34)(17,36,22,38)(21,41,26,44)(27,43,40,32)(28,35,39,42)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346690);