Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1879 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,13)(7,18)(8,21)(9,24)(10,11)(14,30)(15,17)(16,22)(19,23)(20,35)(25,27)(26,28)(29,31)(34,36);(5,6,10,16,12,17,11,27)(7,14,19,32,20,30,28,33)(8,22,24,13,9,25,21,15)(18,29,26,36,35,34,23,31);(5,7)(6,14)(8,23)(9,26)(10,19)(11,28)(12,20)(13,29)(15,34)(16,32)(17,30)(18,24)(21,35)(22,31)(25,36)(27,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=ehi,h^b=hi,d^c=dhi,e^c=eh,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72321310, #72321366, #72321421, … |
| maximal normal subgroup of | #72321310, #72321366, #72321421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53960);