Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | C2×#135 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,14,19,15,18,30,20)(11,16,22,33,23,32,13,28)(12,24,25,31,27,39,26,40)(17,34,36,38,35,29,37,21);(1,2)(3,7)(4,8)(5,6)(9,11,14,22,15,23,30,13)(10,16,19,33,18,32,20,28)(12,21,25,34,27,38,26,29)(17,31,36,39,35,40,37,24);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,21)(13,29)(14,25)(15,27)(16,31)(18,35)(19,36)(20,37)(22,34)(23,38)(24,28)(26,30)(32,40)(33,39);(1,4,5,3)(2,7,6,8)(10,18)(16,32)(17,35)(19,20)(24,39)(28,33)(31,40)(36,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=e,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,h^a=hi,d^b=de,f^c=fh,h^c=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405612);