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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2206 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)(7,10)(8,9)(11,12)(13,14,18,22,19,23,30,17)(15,24,26,36,27,40,34,20)(16,28,33,31,32,21,37,39)(25,41,29,35,42,38,43,44);(2,4)(13,15,19,27)(14,20,23,36)(16,29,32,43)(17,24,22,40)(18,34,30,26)(21,38,28,41)(25,37,42,33)(31,44,39,35);(6,9)(10,12)(13,16,30,37,19,32,18,33)(14,21,17,31,23,28,22,39)(15,25,34,43,27,42,26,29)(20,35,40,41,36,44,24,38);(13,17,30,23,19,22,18,14)(15,20,34,40,27,36,26,24)(16,31,37,28,32,39,33,21)(25,35,43,41,42,44,29,38);(5,7,8,11)(6,10,9,12)(16,32)(21,28)(25,42)(29,43)(31,39)(33,37)(35,44)(38,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=gi,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, 6374718);