Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3864 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)(5,6,8,9)(7,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,22,24,26)(23,25,27,28);(6,9)(10,11)(21,23)(22,25)(24,27)(26,28);(5,7)(6,10)(8,12)(9,11)(14,17)(18,20)(23,27)(25,28);(2,4)(7,12)(10,11);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|a^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279965, #72280049, #72280122, … |
| maximal normal subgroup of | #72279965, #72280049, #72280122, … |
| maximal quotient of | #72336042 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518230);