Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
4940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(6,8)(13,14)(15,18)(16,17)(19,20);(2,4)(10,12)(14,17)(18,20);(1,2)(3,4)(9,10)(11,12)(15,19)(18,20);(5,6)(7,8)(13,14)(15,20)(16,17)(18,19);(1,2)(3,4)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^9|b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72297155, #72297163, #72297199, … |
| maximal normal subgroup of | #72297155, #72297163, #72297199, … |
| maximal quotient of | #72298462, #72330337 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3941033);