Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140370 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1943 |
| 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)(6,12,15,28)(7,17)(8,21)(9,10)(13,18,31,35)(14,29,16,27)(19,36)(20,25)(23,24)(26,32,33,30);(1,2,3,4)(5,6,8,12,10,16,21,29,11,15,24,28,9,14,23,27)(7,18,19,30,17,33,34,31,20,35,36,32,25,26,22,13);(5,7,9,17,11,20,10,25)(6,13,14,30,15,31,16,32)(8,22,23,19,24,34,21,36)(12,26,27,18,28,33,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=e,d^2=h,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,h^a=hi,c^b=cf,g^b=gi,d^c=dhi,f^c=fhi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239802, #72239804, #72239822, … |
| maximal normal subgroup of | #72239802, #72239804, #72239822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56439);