Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3345 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,31)(28,30)(29,32);(2,3)(4,8)(10,13)(14,16)(17,18,20,22)(19,21,24,23)(25,27,30,32)(26,29,28,31);(2,3)(6,7)(11,15)(14,16)(19,24)(21,23)(26,28)(27,32);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)(25,28,30,26)(27,29,32,31);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,29)(26,27)(28,32)(30,31)Polycyclic presentation:
2^9|b^2=ghi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
| maximal subgroup of | #72326584, #72326603, #72326638 |
| maximal normal subgroup of | #72326584, #72326603, #72326638 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4381988);