Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347305 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3490 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,10)(3,12,13,25)(4,16,19,9)(6,11,20,8)(7,17,22,14)(15,27,29,28)(18,24,32,26)(21,30,23,31);(1,3)(2,7)(4,17)(5,13)(6,15)(8,21)(9,25)(10,22)(11,23)(12,16)(14,19)(18,31)(20,29)(24,27)(26,28)(30,32);(4,18)(9,26)(12,27)(14,30)(16,24)(17,31)(19,32)(25,28);(2,8)(7,21)(9,24)(10,11)(12,28)(16,26)(22,23)(25,27);(3,13)(7,22)(12,25)(14,17)(15,29)(21,23)(27,28)(30,31)Polycyclic presentation:
2^9|a^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232843, #72233045, #72234669, … |
| maximal normal subgroup of | #72232843, #72233045, #72234669, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254502);