Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,12,13)(11,14,16,15)(17,18)(19,23)(20,21)(22,24)(25,26,29,32)(27,31,28,30);(1,2,4,6)(3,5,8,7)(10,13)(11,16)(18,21)(22,23)(25,27)(26,30)(28,29)(31,32);(1,2)(3,5)(4,6)(7,8)(9,11,12,16)(10,14,13,15)(17,18,20,21)(19,22,24,23)(25,26,29,32)(27,30,28,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,24)(18,22,21,23)(25,28,29,27)(26,31,32,30);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,26)(27,31)(28,30)(29,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=f,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232640, #72232652, #72232663, … |
| maximal normal subgroup of | #72232640, #72232652, #72232663, … |
| maximal quotient of | #72329054, #72329057 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3790232);