Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,21,17,20,22,15)(16,24,26,23,25,19,27,28);(1,2,3,4)(6,9)(10,11)(13,15,22,20,17,21,18,14)(16,23,27,24,25,28,26,19);(5,7,8,12)(6,10,9,11)(13,16,17,25)(14,19,20,24)(15,23,21,28)(18,26,22,27);(16,25)(19,24)(23,28)(26,27);(14,20)(15,21)(19,24)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224722, #72315249, #72315253 |
| maximal normal subgroup of | #72224722, #72315249, #72315253 |
| maximal quotient of | #72316241 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279371);