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 | |
4712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,18)(15,16)(19,20);(6,9)(10,12);(13,14)(15,18)(16,20)(17,19);(5,7,8,11)(6,10,9,12);(13,15,17,16)(14,18,19,20)Polycyclic presentation:
2^9|a^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,g^c=gi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216622, #72305410, #117×AGL(1,ℤ/8), … |
| maximal normal subgroup of | #72216622, #72305410, #117×AGL(1,ℤ/8), … |
| maximal quotient of | #72216479, #72306164 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278176);