Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,4,7,6)(10,14)(11,15)(17,18)(19,20)(21,22)(23,24)(26,29)(30,32);(1,3)(2,6)(4,8)(5,7)(9,10,13,14)(11,12,15,16)(18,22)(19,24)(27,31)(30,32);(17,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,29)(31,32);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,27,28,31)(26,30,29,32);(1,4)(2,7)(3,8)(5,6)(9,12)(10,15)(11,14)(13,16)(17,20)(18,24)(19,22)(21,23)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72260818, #72260822, #72260839 |
| maximal normal subgroup of | #72260818, #72260822, #72260839 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2530442);