Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2892 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)(10,13)(11,15)(17,18)(19,22)(20,21)(23,24)(25,26)(27,28)(29,31)(30,32);(2,3)(4,8)(9,10)(11,16)(12,13)(14,15)(17,18,20,21)(19,23,24,22)(25,27,29,32)(26,30,31,28);(2,3)(6,7)(10,13)(14,16)(18,21)(22,23)(26,31)(28,30);(1,3,5,2)(4,6,8,7)(9,10)(11,16)(12,13)(14,15)(17,18)(19,23)(20,21)(22,24)(25,28,29,30)(26,27,31,32);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|b^2=fi,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72240130, #72240183, #72240185, … |
| maximal normal subgroup of | #72240130, #72240183, #72240185, … |
| maximal quotient of | #72251695 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2997107);