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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2828 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,20)(19,24,21,23)(25,26)(27,28)(29,32)(30,31);(2,5)(6,7)(9,11,14,12)(10,13,16,15)(17,19)(18,23)(20,24)(21,22)(25,27,31,32)(26,29,30,28);(9,11,14,12)(10,15,16,13)(17,18,22,20)(19,24,21,23)(25,28,31,29)(26,32,30,27);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,15,16,13)(17,20,22,18)(19,24,21,23)(25,29,31,28)(26,32,30,27);(1,2)(3,7)(4,5)(6,8)(9,13,14,15)(10,12,16,11)(17,21,22,19)(18,23,20,24)(25,30,31,26)(27,29,32,28)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=ghi,d^2=f,e^2=ghi,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2958022);