Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | #15947 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,4)(10,13)(14,16)(18,20)(22,23)(25,26)(27,30)(28,32)(29,31);(1,2)(3,7)(4,8)(5,6)(11,15)(14,16)(17,18)(19,23)(20,21)(22,24)(25,27,29,32)(26,30,31,28);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,20,23)(25,28)(26,27)(29,30)(31,32);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)(25,27)(26,28)(29,32)(30,31);(1,2)(3,8)(4,7)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,22)(21,24)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72300734, #72300743, #72300802, … |
| maximal normal subgroup of | #72300734, #72300743, #72300802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3408503);