Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9938 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,8,11,12)(9,10)(13,14,17,22,18,20,16,21)(15,23,25,19,26,28,24,27);(2,4)(5,7)(6,8)(9,11)(10,12)(13,15,18,26)(14,19,20,27)(16,25,17,24)(21,28,22,23);(5,8)(6,7)(9,12)(10,11)(14,20)(16,17)(23,28)(24,25)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dhi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284120, #72284235, #72284659, … |
| maximal normal subgroup of | #72284120, #72284235, #72284659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19781);