Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C23×C42 |
| socle | C23 | C24×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,21,18,20,17,22)(15,19,23,27,25,26,24,28);(6,9)(7,11)(13,15,17,24,18,25,16,23)(14,19,22,28,20,26,21,27);(5,7,8,11)(6,10,9,12);(6,9)(10,12);(14,20)(19,26)(21,22)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=f,g^2=i,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf176 conjugacy classes — view the full character table.
| maximal subgroup of | #72311780, #72311798, #72314658, … |
| maximal normal subgroup of | #72311780, #72311798, #72314658, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277429);