Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2436 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)(7,9)(8,10)(11,12)(13,14)(15,21)(16,20)(17,23)(18,24)(19,22)(25,27)(26,28);(5,7)(6,9)(8,11)(10,12)(13,15,17,16,19,26,18,25)(14,20,24,21,22,27,23,28);(2,4)(7,11)(9,12)(13,16,17,26,19,25,18,15)(14,21,24,27,22,28,23,20);(6,10)(9,12)(14,22)(20,27)(21,28)(23,24);(13,17,19,18)(14,23,22,24)(15,16,26,25)(20,28,27,21)Polycyclic presentation:
2^9|b^2=f,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72311041, #72311048, #72311050, … |
| maximal normal subgroup of | #72311041, #72311048, #72311050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340444);