Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,31)(17,23)(18,24)(19,25)(21,33)(22,32)(26,41)(27,44)(28,36)(29,37)(30,38)(34,40)(35,43)(39,42);(6,9)(10,12)(13,15)(14,20)(16,27)(17,28)(18,29)(19,30)(21,39)(22,35)(23,36)(24,37)(25,38)(26,34)(31,44)(32,43)(33,42)(40,41);(2,4)(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,26)(17,32)(18,33)(19,34)(20,35)(22,25)(23,31)(24,40)(27,30)(28,42)(29,43)(36,41)(37,44)(38,39);(13,16)(14,22)(15,27)(17,32)(18,33)(19,34)(20,35)(21,25)(23,40)(24,31)(26,30)(28,43)(29,42)(36,41)(37,44)(38,39);(13,17,19,18)(14,23,25,24)(15,28,30,29)(16,32,34,33)(20,36,38,37)(21,31,22,40)(26,42,27,43)(35,41,39,44)Polycyclic presentation:
2^9|c^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,c^b=ci,g^c=gi,h^c=hi,g^d=gi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536412);