Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C22×D8 | C23 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2)(3,4)(5,6,9,12)(7,11,8,10)(13,14,16,21)(15,24,26,19)(17,20,29,36)(18,32,30,23)(22,38,37,31)(25,41,42,34)(27,44,43,39)(28,35,33,40);(5,7,9,8)(6,10,12,11)(13,15,16,26)(14,19,21,24)(17,25,29,42)(18,33,30,28)(20,34,36,41)(22,39,37,44)(23,40,32,35)(27,38,43,31);(2,4)(14,20)(15,25)(17,31)(18,30)(21,36)(22,23)(26,42)(27,28)(29,38)(32,37)(33,43)(34,39)(35,40)(41,44);(5,8,9,7)(6,11,12,10)(14,21)(19,24)(20,36)(22,37)(23,32)(34,41)(35,40)(39,44)Polycyclic presentation:
2^9|a^2=e,b^2=e,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,f^c=fh,g^c=gh,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408100);