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 | |
518 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,11,14,9,17,25,33,12,16,24,32,10,15,8,23)(7,18,21,35,20,13,27,31,22,29,36,34,26,30,19,28);(1,2,4,3)(5,7,9,20,12,22,10,26)(6,13,15,18,16,30,17,29)(8,19,11,21,25,27,24,36)(14,31,23,35,32,28,33,34);(1,3,4,2)(5,8,9,11,12,25,10,24)(6,14,15,23,16,32,17,33)(7,19,20,21,22,27,26,36)(13,28,18,34,30,31,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239414, #72239421, #72239426, … |
| maximal normal subgroup of | #72239414, #72239421, #72239426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55873);