Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | #33579 | C2 |
| 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 | |
561 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)(9,10,14,19,15,21,31,20)(11,16,13,29,23,18,32,34)(12,24,26,36,28,40,27,33)(17,35,37,22,38,30,39,25);(1,2,5,6)(3,7,4,8)(9,11,14,13,15,23,31,32)(10,16,19,29,21,18,20,34)(12,25,26,35,28,22,27,30)(17,36,37,40,38,33,39,24);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,22)(13,30)(14,26)(15,28)(16,33)(18,36)(19,37)(20,39)(21,38)(23,25)(24,29)(27,31)(32,35)(34,40);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=e,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,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, 405650);