Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3227 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #24299 |
| prime radical | #3054 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,16)(4,20,23,30)(5,9)(6,15)(7,12)(8,14)(10,21,28,32)(11,19)(13,18)(17,29,31,27)(22,26,24,25);(1,3,6,5,8,19,7,18)(2,9,12,16,14,13,15,11)(4,17,22,21,23,31,24,32)(10,26,27,20,28,25,29,30);(1,4)(2,10)(3,17)(5,21)(6,22)(7,24)(8,23)(9,25)(11,20)(12,27)(13,26)(14,28)(15,29)(16,30)(18,32)(19,31);(1,5,6,19,8,18,7,3)(2,11,12,9,14,16,15,13)(4,21,22,31,23,32,24,17)(10,20,27,25,28,30,29,26)Polycyclic presentation:
2^9|a^2=eh,b^2=e,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,f^c=fhi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72219574, #72219575, #72219578, … |
| maximal normal subgroup of | #72219574, #72219575, #72219578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408089);