Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,21)(19,24,20,23)(26,28)(27,31);(2,4)(6,8)(9,10,12,14)(11,15,16,13)(17,19,22,20)(18,23,21,24)(25,26)(27,32)(28,29)(30,31);(2,4)(6,8)(11,16)(13,15)(18,21)(23,24)(26,28)(30,32);(1,3,5,7)(2,6,4,8)(9,10)(11,15)(12,14)(13,16)(17,20,22,19)(18,24,21,23)(25,27,29,31)(26,30,28,32);(1,4,5,2)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,21,22,18)(19,23,20,24)(25,28,29,26)(27,30,31,32)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72326742, #72326744, #72326779, … |
| maximal normal subgroup of | #72326742, #72326744, #72326779, … |
| maximal quotient of | #72328621, #72331595, #72331605 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3790299);