Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(11,15)(14,16)(17,18,20,22)(19,23,24,21)(25,26)(27,30)(28,33)(29,32)(31,35)(34,37)(36,40)(38,39);(2,3)(6,7)(17,18,20,22)(19,21,24,23)(26,27)(28,35)(29,37)(39,40);(1,2,5,3)(4,6,8,7)(9,10,13,12)(11,14,15,16)(17,19)(18,21)(20,24)(22,23)(25,27,30,26)(28,31,35,33)(29,34,37,32)(36,39,38,40);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,14,12,16)(17,18)(19,21)(20,22)(23,24)(25,28,30,35)(26,31,27,33)(29,38,37,36)(32,40,34,39);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,19)(18,21)(20,24)(22,23)(25,29,30,37)(26,32,27,34)(28,36,35,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=ghi,d^2=ghi,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3239881);