Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(14,17)(15,19)(21,22)(23,25)(24,26)(27,28);(6,9)(10,11)(13,14)(15,20)(16,17)(18,19)(21,22,24,26)(23,25,27,28);(1,2)(3,4)(7,12)(10,11)(14,17)(18,20)(23,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,25)(24,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|b^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72234678, #72234679, #72234682, … |
| maximal normal subgroup of | #72234678, #72234679, #72234682, … |
| maximal quotient of | #72328394, #72328824, #72329035 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4227183);