Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3150 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,19,23)(15,24,26,39)(16,21,30,37)(17,22,18,32)(20,27,34,28)(25,38,40,44)(29,35,31,36)(33,42,43,41);(5,6)(7,10)(8,9)(11,12)(13,15,18,28,19,26,17,27)(14,20,22,24,23,34,32,39)(16,25,29,41,30,40,31,42)(21,33,36,38,37,43,35,44);(1,2,3,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,31)(18,29)(19,30)(20,33)(22,36)(23,37)(24,38)(26,40)(27,42)(28,41)(32,35)(34,43)(39,44);(5,7,8,11)(6,10,9,12)(15,26)(20,34)(24,39)(25,40)(27,28)(33,43)(38,44)(41,42);(13,17,19,18)(14,22,23,32)(15,27,26,28)(16,29,30,31)(20,24,34,39)(21,35,37,36)(25,41,40,42)(33,44,43,38)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^b=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343165);