Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480454 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35088 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2143 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,13,22,23)(7,15,18,29,17,11,21,25)(8,12,19,27,20,26,33,34)(16,28,31,36,30,24,32,35);(5,7,9,21)(6,11,23,29)(8,16,19,32)(10,17,22,18)(12,24,34,36)(13,15,14,25)(20,30,33,31)(26,28,27,35);(2,4)(5,8,10,20)(6,12,13,26)(7,16,17,30)(9,19,22,33)(11,24,15,28)(14,27,23,34)(18,31,21,32)(25,35,29,36);(6,13)(11,15)(12,26)(14,23)(24,28)(25,29)(27,34)(35,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279131, #72279149, #72279206, … |
| maximal normal subgroup of | #72279131, #72279149, #72279206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401892);