Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,13,14)(11,12,15,16)(17,18)(19,23)(20,24)(21,22)(25,26,29,31)(27,28,32,30);(3,7)(6,8)(9,11,13,15)(10,12,14,16)(17,19)(18,20)(21,24)(22,23)(25,26,29,31)(27,28,32,30);(2,5)(6,8)(10,14)(11,15)(17,18,21,22)(19,20,24,23)(25,27,29,32)(26,30,31,28);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,27)(26,28)(29,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72336512, #72336514, #72336531, … |
| maximal normal subgroup of | #72336512, #72336514, #72336531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5398980);