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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2678 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)(18,20)(22,23)(25,26,29,28)(27,31,32,30);(2,4)(6,8)(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,21)(22,24)(25,26)(27,30)(28,29)(31,32);(2,4)(6,8)(11,16)(13,15)(18,20)(22,23)(26,28)(30,31);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,24)(18,22,20,23)(25,27,29,32)(26,30,28,31);(1,4,5,2)(3,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,21,18)(19,22,24,23)(25,28,29,26)(27,30,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=fghi,d^2=h,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 | #72326697, #72326716, #72326766, … |
| maximal normal subgroup of | #72326697, #72326716, #72326766, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3792091);