Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7796 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2,3,4)(6,9)(7,11)(13,14,19,23)(15,20,24,33)(16,28,29,21)(17,18,22,32)(25,39,37,34)(26,27,35,38)(30,42,36,31)(40,44,43,41);(1,2)(3,4)(6,9)(10,12)(15,24)(16,29)(20,33)(21,28)(26,35)(27,38)(30,36)(31,42);(5,6)(7,10)(8,9)(11,12)(13,15,19,24)(14,20,23,33)(16,25,29,37)(17,26,22,35)(18,27,32,38)(21,34,28,39)(30,40,36,43)(31,41,42,44);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,25)(17,30)(18,31)(19,29)(20,34)(22,36)(23,28)(24,37)(26,40)(27,41)(32,42)(33,39)(35,43)(38,44);(13,17)(14,18)(15,26)(16,30)(19,22)(20,27)(21,31)(23,32)(24,35)(25,40)(28,42)(29,36)(33,38)(34,41)(37,43)(39,44);(13,18)(14,22)(15,27)(16,31)(17,23)(19,32)(20,35)(21,36)(24,38)(25,41)(26,33)(28,30)(29,42)(34,43)(37,44)(39,40)Polycyclic presentation:
2^9|a^2=ghi,c^2=hi,d^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566243);