Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12741 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| 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 | |
1046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,16,11,17,10,27)(7,18,20,34,21,29,28,33)(8,22,25,15,26,13,24,14)(12,23,30,35,32,36,31,19);(1,2,3,4)(5,7)(6,12)(8,23)(9,20)(10,28)(11,21)(13,29)(14,33)(15,34)(16,30)(17,32)(18,22)(19,24)(25,35)(26,36)(27,31);(6,13)(8,24)(9,10)(12,29)(14,27)(15,16)(17,22)(18,32)(19,23)(20,28)(25,26)(30,34)(31,33)(35,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dh,e^b=eh,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249946, #72249959, #72249979, … |
| maximal normal subgroup of | #72249946, #72249959, #72249979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53930);