Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14762 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,15,16,14)(18,22)(19,23)(26,27)(28,32);(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,24,23,20)(26,27)(30,31);(10,13)(14,15)(18,22)(20,24)(25,26,29,27)(28,30,32,31);(1,2)(3,5)(4,6)(7,8)(9,11,12,16)(10,14,13,15)(17,19,21,23)(18,20,22,24)(25,27,29,26)(28,31,32,30);(1,3,4,7)(2,5,6,8)(9,10)(11,15)(12,13)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=fh,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72304074, #72304081, #72304136 |
| maximal normal subgroup of | #72304074, #72304081, #72304136 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3566670);