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) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(18,19)(20,24)(26,29)(30,31);(2,3)(6,7)(9,10)(11,14)(12,13)(15,16)(18,19)(22,23)(25,26,28,29)(27,31,32,30);(4,8)(6,7)(10,13)(14,15)(17,18,21,19)(20,22,24,23)(25,27,28,32)(26,30,29,31);(1,3,5,2)(4,6,8,7)(9,11,12,16)(10,14,13,15)(17,19,21,18)(20,23,24,22)(25,26)(27,30)(28,29)(31,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,31)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=fghi,d^2=ghi,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 | #72327630, #72327632, #72327775 |
| maximal normal subgroup of | #72327630, #72327632, #72327775 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3461326);