Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190022 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,13,15,10)(8,16)(9,12)(11,17)(14,18,20,19);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20);(5,8)(6,11,12,16)(7,14)(9,17)(10,18,13,19)(15,20);(2,4)Polycyclic presentation:
2^9|a^2=h,c^2=e,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219301, #72219304, #72219305, … |
| maximal normal subgroup of | #72219301, #72219304, #72219305, … |
| maximal quotient of | #72234592, #72259052, #72265195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299427);