Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
2132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10)(11,14)(12,13)(15,16)(18,22)(23,24)(25,26,29,32)(27,31,28,30);(2,5)(6,8)(10,13)(11,15)(17,18,21,22)(19,23,20,24)(25,26)(27,30)(28,31)(29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,13)(14,15)(17,18)(19,24)(20,23)(21,22)(25,26)(27,31)(28,30)(29,32);(17,19,21,20)(18,23,22,24)(25,27,29,28)(26,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=i,b^2=gi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72258895, #72258905 |
| maximal normal subgroup of | #72258895, #72258905 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393533);