Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×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×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3050 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)(5,6,8,9)(7,11,12,10)(13,14,18,24)(15,26,30,38)(16,22,35,21)(17,25,19,23)(20,31,37,29)(27,42,28,39)(32,44,43,36)(33,40,34,41);(1,2,3,4)(6,9)(10,11)(13,15,19,31,18,30,17,29)(14,20,25,38,24,37,23,26)(16,32,33,28,35,43,34,27)(21,39,40,44,22,42,41,36);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,27)(17,33)(18,35)(19,34)(20,36)(22,24)(23,40)(25,41)(26,39)(28,30)(29,32)(31,43)(37,44)(38,42);(13,16)(14,22)(15,28)(17,33)(18,35)(19,34)(20,36)(21,24)(23,41)(25,40)(26,39)(27,30)(29,43)(31,32)(37,44)(38,42);(13,17,18,19)(14,23,24,25)(15,29,30,31)(16,33,35,34)(20,26,37,38)(21,40,22,41)(27,32,28,43)(36,39,44,42)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=gi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354884);