Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2557 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)(3,6)(4,7)(5,8)(10,13)(11,15)(17,18)(19,23)(20,22)(21,24)(26,29)(27,31);(1,3,4,8)(2,5,7,6)(10,13)(14,16)(17,19,21,20)(18,22,24,23)(27,31)(30,32);(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,23,24,22)(25,26)(27,30)(28,29)(31,32);(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,23,24,22)(25,27,28,31)(26,30,29,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72303399, #72303510, #72303520, … |
| maximal normal subgroup of | #72303399, #72303510, #72303520, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2853099);