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 | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2104 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)(5,6,9,14,10,13,23,24)(7,15,18,30,17,11,22,26)(8,19,20,33,21,12,27,28)(16,25,32,35,31,29,34,36);(5,7,9,22)(6,11,24,30)(8,16,20,34)(10,17,23,18)(12,25,33,36)(13,15,14,26)(19,29,28,35)(21,31,27,32);(2,4);(5,8,10,21)(6,12,13,19)(7,16,17,31)(9,20,23,27)(11,25,15,29)(14,28,24,33)(18,32,22,34)(26,35,30,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^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 | #72279120, #72279149, #72279243, … |
| maximal normal subgroup of | #72279120, #72279149, #72279243, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401893);