Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #32880 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4493 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)(15,23)(16,21)(17,22)(18,20)(19,25)(24,27)(26,28);(6,9)(10,12)(13,15,17,25)(14,19,22,23)(16,24,18,26)(20,27,21,28);(14,20)(16,18)(19,27)(21,22)(23,28)(24,26);(2,4);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,21,22,20)(15,24,25,26)(19,28,23,27)Polycyclic presentation:
2^9|b^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72256938, #72256952, #72257015, … |
| maximal normal subgroup of | #72256938, #72256952, #72257015, … |
| maximal quotient of | #72264442, #72264977, #72265165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536481);