Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,11,16,27,17)(7,18,21,33,20,34,26,28)(8,13,9,23,22,14,25,32)(12,24,30,19,29,36,31,35);(1,2,3,4)(5,7,11,20)(6,12,16,29)(8,19,22,35)(9,24,25,36)(10,26,27,21)(13,28,14,33)(15,31,17,30)(18,23,34,32);(5,8,10,9,11,22,27,25)(6,13,15,23,16,14,17,32)(7,19,26,24,20,35,21,36)(12,28,31,34,29,33,30,18)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dhi,e^b=ei,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54653);