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×C42 |
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 | |
3672 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)(9,10,13,14)(11,15,16,12)(18,21)(19,23)(25,26)(27,31)(28,30)(29,32);(2,5)(6,8)(9,11,13,16)(10,12,14,15)(17,18)(19,22)(20,21)(23,24)(25,27,30,29)(26,31,28,32);(3,7)(6,8)(11,16)(12,15)(17,18)(19,24)(20,21)(22,23)(25,26,30,28)(27,32,29,31);(1,2)(3,8)(4,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,21)(22,23)(25,28,30,26)(27,31,29,32);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,19,20,23)(18,22,21,24)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=ghi,d^2=ghi,e^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 | #72232606, #72232615, #72232640, … |
| maximal normal subgroup of | #72232606, #72232615, #72232640, … |
| maximal quotient of | #72328844, #72328909, #72330016 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3764121);