Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2274 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,20,24,22)(25,26,29,30)(27,32,28,31);(2,3)(6,7)(10,12)(11,15)(17,19)(18,22)(20,23)(21,24)(26,30)(27,28);(4,8)(6,7)(10,12)(14,16)(18,23)(19,24)(25,27,29,28)(26,31,30,32);(1,3,5,2)(4,6,8,7)(9,11)(10,14)(12,16)(13,15)(17,20,21,22)(18,24,23,19)(25,28,29,27)(26,32,30,31);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,14,15,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,32)(28,31)(29,30)Polycyclic presentation:
2^9|a^2=h,c^2=fgi,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72304068, #72304079, #72304136 |
| maximal normal subgroup of | #72304068, #72304079, #72304136 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3987049);