Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1985 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
490 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,11,14,12,17,24,32,9,16,25,31,10,15,8,23)(7,18,21,30,20,33,19,34,22,13,27,28,26,29,35,36);(1,2)(3,4)(5,7,12,20,9,22,10,26)(6,13,15,33,16,18,17,29)(8,19,11,27,24,35,25,21)(14,30,23,36,31,28,32,34);(2,4)(5,8,12,11,9,24,10,25)(6,14,15,23,16,31,17,32)(7,19,20,27,22,35,26,21)(13,28,33,34,18,30,29,36)Polycyclic presentation:
2^9|a^2=g,b^2=de,c^2=de,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239415, #72239421, #72239431, … |
| maximal normal subgroup of | #72239415, #72239421, #72239431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56497);