Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,16,12,15,11,8)(7,17,20,34,21,33,28,31)(9,23,25,32,27,22,26,14)(13,24,30,35,29,36,18,19);(1,2,3,4)(5,7)(6,13)(8,18)(9,24)(10,20)(11,28)(12,21)(14,31)(15,29)(16,30)(17,23)(19,26)(22,33)(25,35)(27,36)(32,34);(5,8,10,15,12,16,11,6)(7,18,20,29,21,30,28,13)(9,22,25,32,27,23,26,14)(17,19,31,24,33,35,34,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,d^b=dh,h^b=hi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250147, #72250150, #72250158, … |
| maximal normal subgroup of | #72250147, #72250150, #72250158, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54676);