Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13522 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2088 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #137 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
456 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)(5,6,11,16,28,15,9,14,12,17,29,33,10,7,18,20)(8,19,25,36,23,13,22,31,27,34,24,21,26,32,35,30);(1,2,4,3)(5,7,10,6,12,15,28,17)(8,21,23,31,27,36,26,30)(9,20,11,16,18,14,29,33)(13,25,19,22,32,24,34,35);(1,3,4,2)(5,8,9,22,10,26,11,25,12,27,18,35,28,23,29,24)(6,13,14,21,7,19,16,31,17,32,20,36,15,34,33,30)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=dh,f^b=fhi,g^b=ghi,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250752, #72250764, #72250785, … |
| maximal normal subgroup of | #72250752, #72250764, #72250785, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56248);