Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#540 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3750 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)(15,17)(16,20)(18,23)(19,21)(22,25)(24,27)(26,28);(2,4)(13,15,17,14,19,23,18,21)(16,22,25,20,24,28,26,27);(6,9)(10,12)(13,16,19,24)(14,20,21,27)(15,22,23,28)(17,25,18,26);(5,7,8,11)(6,10,9,12)(16,24)(20,27)(22,28)(25,26);(13,17,19,18)(14,15,21,23)(16,25,24,26)(20,22,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72292819, #72292820, #72292824, … |
| maximal normal subgroup of | #72292819, #72292820, #72292824, … |
| maximal quotient of | #72293232, #72316287, #72316288 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279643);