Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4544 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)(15,22)(16,21)(17,25)(18,20)(19,23)(24,28)(26,27);(6,9)(10,12);(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,28)(22,25);(5,7,8,11)(6,10,9,12)(14,20)(19,27)(21,25)(22,28);(13,16,18,17)(14,21,20,25)(15,23,24,26)(19,28,27,22)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,g^c=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72257602, #72257691, #72257699, … |
| maximal normal subgroup of | #72257602, #72257691, #72257699, … |
| maximal quotient of | #72257789, #72264239, #72304902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278168);