Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442597 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2487 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,10,6,11,5,18)(3,12,19,22,13,26,15,8)(4,9,20,24,16,23,17,29)(14,25,30,31,27,32,28,21);(3,13)(4,16)(15,19)(17,20);(1,3,5,15,6,13,7,19)(2,8,10,12,11,22,18,26)(4,14,17,28,16,27,20,30)(9,21,24,25,23,31,29,32);(1,4)(2,9)(3,14)(5,17)(6,16)(7,20)(8,21)(10,24)(11,23)(12,25)(13,27)(15,28)(18,29)(19,30)(22,31)(26,32);(3,13)(8,22)(12,26)(14,27)(15,19)(21,31)(25,32)(28,30)Polycyclic presentation:
2^9|a^2=gh,c^2=g,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285642, #72285784, #72286225, … |
| maximal normal subgroup of | #72285642, #72285784, #72286225, … |
| maximal quotient of | C2×#6458220 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365416);