Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #239359 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#394 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
975 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,22,26)(19,27,29,31)(20,23,21,25)(24,28,32,30);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(18,23)(19,28)(20,21)(24,32)(25,26)(29,30);(6,12)(13,16)(17,19,20,30,22,29,21,28)(18,24,25,27,26,32,23,31);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(19,29)(24,32)(27,31)(28,30)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=hi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gh,d^c=dh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72268384, #72268387, #72268408, … |
| maximal normal subgroup of | #72268384, #72268387, #72268408, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 146561);