Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C2×C4 | C82 |
| Frattini, cosocle | C42×C8 | C22 |
| socle, prime radical | C23 | #336 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 40 points:
(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22)(25,26,27,31,29,32,34,38,28,30,33,36,35,37,39,40);(1,2,3,5,4,6,7,8)(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24)(26,30)(27,33)(32,37)(34,39)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=h,g^2=i,b^a=bc,c^a=ci,e^a=efi,d^b=df128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#93250, #72210670, #72210671, … |
| maximal normal subgroup of | C2×#93250, #72210670, #72210671, … |
| maximal quotient of | #72210244, #72210248, #72210250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 446);