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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10,14,16)(11,13,12,15)(18,19)(20,24)(28,32)(30,31);(1,2,4,5)(3,8,7,6)(9,11,14,12)(10,15,16,13)(18,19)(22,23)(25,26,29,27)(28,30,32,31);(3,7)(6,8)(10,16)(13,15)(17,18,21,19)(20,22,24,23)(25,27,29,26)(28,31,32,30);(1,2)(3,8)(4,5)(6,7)(9,12,14,11)(10,13,16,15)(17,19,21,18)(20,23,24,22)(25,28)(26,30)(27,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,13)(10,12)(11,16)(14,15)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=fi,d^2=g,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3850477);