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 | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,17,13,18,12,16)(7,19,21,11,22,29,15,9)(8,23,26,14,27,24,31,33)(20,30,36,25,32,28,34,35);(1,2,3,4)(5,7,9,18,12,21,29,16,13,22,11,6,10,15,19,17)(8,24,25,36,26,14,28,20,27,23,35,34,31,33,30,32);(5,8,10,26,13,27,12,31)(6,14,16,23,18,33,17,24)(7,20,15,34,22,32,21,36)(9,28,19,35,11,30,29,25)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dh,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250231, #72250234, #72250256, … |
| maximal normal subgroup of | #72250231, #72250234, #72250256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56437);