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) = C25×C4 |
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 | |
3022 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)(3,7)(4,5)(6,8)(10,13)(11,15)(18,20)(22,23)(26,27)(28,32);(2,5)(3,8)(9,10,12,13)(11,16,15,14)(17,18)(19,23)(20,21)(22,24)(26,27)(30,31);(3,8)(6,7)(11,15)(14,16)(18,20)(22,23)(25,26,29,27)(28,30,32,31);(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,13)(14,15)(17,19,21,24)(18,22,20,23)(25,27,29,26)(28,31,32,30);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,22,24,23)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|b^2=gi,c^2=g,d^2=h,e^2=fg,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 | #72300906, #72300908, #72300928, … |
| maximal normal subgroup of | #72300906, #72300908, #72300928, … |
| maximal quotient of | #72330985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3790649);