Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500450 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2040 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)(13,14)(15,24)(16,21)(17,22)(18,33)(19,23)(20,27)(25,41)(26,42)(28,36)(29,38)(30,40)(31,37)(32,39)(34,44)(35,43);(5,6)(7,11)(8,10)(9,12)(13,15,17,31,19,28,18,27)(14,20,33,36,23,37,22,24)(16,26,32,44,30,25,29,43)(21,35,38,41,40,34,39,42);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,30)(14,21,23,40)(15,25,28,26)(17,32,18,29)(20,34,37,35)(22,39,33,38)(24,41,36,42)(27,44,31,43);(7,12)(9,11)(13,16)(14,21)(15,26)(17,32)(18,29)(19,30)(20,35)(22,39)(23,40)(24,42)(25,28)(27,43)(31,44)(33,38)(34,37)(36,41);(6,10)(9,11)(13,17,19,18)(14,22,23,33)(15,27,28,31)(16,29,30,32)(20,36,37,24)(21,38,40,39)(25,43,26,44)(34,41,35,42)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6410120);