Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10456 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#394 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1049 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,13)(5,9)(6,15)(7,10)(8,12)(14,16)(17,18)(19,27)(20,23)(21,25)(22,26)(24,30)(28,31)(29,32);(1,3)(2,8)(4,9)(5,12)(6,16)(7,13)(10,14)(11,15)(18,23)(19,28)(20,21)(24,27)(25,26)(31,32);(1,4,6,8,7,11,5,14)(2,3,9,12,10,13,15,16)(17,19,20,29,22,28,21,30)(18,24,25,31,26,32,23,27);(1,5,7,6)(2,9,10,15)(3,12,13,16)(4,14,11,8)Polycyclic presentation:
2^9|c^2=e,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,f^c=fhi,g^c=gh80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#240064, #72213461, #72216075, … |
| maximal normal subgroup of | C2×#240064, #72213461, #72216075, … |
| maximal quotient of | #72213386, C2×#240064 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147260);