Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1884 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)(7,11)(8,12)(9,10)(13,14)(15,24)(16,21)(17,22)(18,23)(19,27)(20,31)(25,35)(26,43)(28,42)(29,37)(30,40)(32,41)(33,36)(34,38)(39,44);(6,10)(11,12)(13,15)(14,19)(16,29)(17,27)(18,33)(20,37)(21,39)(22,24)(23,42)(25,30)(26,34)(28,32)(31,44)(35,38)(36,41)(40,43);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,17,31)(14,20,22,21)(15,25,27,26)(18,34,32,30)(19,35,24,43)(23,40,41,38)(28,44,33,29)(36,39,42,37);(5,8,9,7)(6,12,10,11)(16,30)(18,32)(20,38)(21,40)(23,41)(25,29)(26,44)(28,33)(31,34)(35,37)(36,42)(39,43);(13,16,17,31)(14,21,22,20)(15,26,27,25)(18,34,32,30)(19,35,24,43)(23,38,41,40)(28,29,33,44)(36,39,42,37)Polycyclic presentation:
2^9|c^2=fgi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6492065);