Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433049 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1968 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)(7,11)(10,12)(13,14)(15,24)(16,29)(17,22)(18,33)(19,23)(20,28)(21,31)(25,40)(26,39)(27,35)(30,37)(32,36)(34,43)(38,41)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,28,19,26,17,27)(14,20,22,24,23,35,33,39)(16,25,30,42,31,41,32,43)(21,34,37,38,29,44,36,40);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,32)(18,30)(19,31)(20,34)(22,37)(23,29)(24,38)(26,41)(27,43)(28,42)(33,36)(35,44)(39,40);(5,7,8,11)(6,10,9,12)(15,26)(20,35)(24,39)(25,41)(27,28)(34,44)(38,40)(42,43);(13,17,19,18)(14,22,23,33)(15,27,26,28)(16,30,31,32)(20,24,35,39)(21,36,29,37)(25,42,41,43)(34,40,44,38)Polycyclic presentation:
2^9|a^2=gi,b^2=f,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, 6342756);