Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3234 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)(6,9)(7,11)(13,14,18,23,19,24,17,22)(15,20,28,36,26,34,27,35)(16,29,30,21,32,37,31,38)(25,39,41,33,40,44,42,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,35)(23,36)(24,34)(29,39)(30,41)(31,42)(32,40)(37,44)(38,43);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,31)(19,32)(20,33)(22,37)(23,29)(24,38)(26,40)(27,41)(28,42)(34,43)(35,44)(36,39);(5,7,8,11)(6,10,9,12)(15,26)(20,34)(25,40)(27,28)(33,43)(35,36)(39,44)(41,42);(13,17,19,18)(14,22,24,23)(15,27,26,28)(16,30,32,31)(20,35,34,36)(21,37,38,29)(25,41,40,42)(33,44,43,39)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,d^b=di,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281642);