Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2082 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)(8,12)(9,11)(13,14)(15,26)(16,30)(17,32)(18,24)(19,25)(20,23)(21,29)(22,27)(28,40)(31,43)(33,44)(34,39)(35,38)(36,42)(37,41);(5,7,10,6)(8,12,9,11)(13,15)(14,21)(16,20)(17,27)(18,31)(19,29)(22,23)(24,37)(25,26)(28,33)(30,32)(34,41)(35,42)(36,44)(38,40)(39,43);(2,4)(13,16,17,29,19,27,20,15)(14,22,32,26,25,30,23,21)(18,31,33,42,34,41,35,28)(24,37,44,40,39,43,38,36);(5,8,10,9)(6,11,7,12)(13,17,19,20)(14,23,25,32)(15,27,29,16)(18,33,34,35)(21,22,26,30)(24,38,39,44)(28,41,42,31)(36,37,40,43);(5,9,10,8)(6,11,7,12)(13,18,20,35,19,34,17,33)(14,24,23,38,25,39,32,44)(15,28,27,41,29,42,16,31)(21,36,30,43,26,40,22,37)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6718553);