Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2482 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,9,15,11,16,10,14)(7,17,21,12,20,26,19,27)(8,13,23,29,22,30,24,31)(18,33,36,28,34,32,35,25);(1,2,4,3)(5,7)(6,12,16,27)(8,18)(9,19)(10,21)(11,20)(13,28,30,25)(14,26,15,17)(22,34)(23,35)(24,36)(29,33,31,32);(1,3,4,2)(5,8,9,23,11,22,10,24)(6,13,15,29,16,30,14,31)(7,18,19,35,20,34,21,36)(12,25,17,32,27,28,26,33);(6,14,16,15)(8,22)(12,26,27,17)(13,29,30,31)(18,34)(23,24)(25,32,28,33)(35,36)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=e,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72320063 |
| maximal normal subgroup of | #72320063 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318691);