Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470019 |
| 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×#3506 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,21,24)(19,25,27,32)(20,28,30,23)(22,29,31,26);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19)(18,22)(20,29)(21,27)(23,32)(24,31)(25,28)(26,30);(3,9)(6,13)(10,11)(14,15)(20,30)(23,28)(25,32)(26,29);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15);(2,8)(6,13)(7,16)(14,15)Polycyclic presentation:
2^9|a^2=i,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388801);