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) = C27×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 | |
5948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10,13,12)(11,15,16,14)(17,18,22,24)(19,21,20,23);(10,12)(14,15)(17,19,22,20)(18,23,24,21);(5,6)(7,8)(11,16)(14,15)(18,24)(19,20);(1,2)(3,4)(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,20,22,19)(18,23,24,21);(9,12,13,10)(11,15,16,14)(17,21,22,23)(18,19,24,20)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=f,e^2=f,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 | #72232517, #72232532, #72232540, … |
| maximal normal subgroup of | #72232517, #72232532, #72232540, … |
| maximal quotient of | #72251011, #72329984, #72330463 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4219511);