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 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(10,13)(11,15)(19,23)(20,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(11,15)(14,16)(17,18,21,22)(19,24,23,20)(26,30)(28,36)(31,33)(34,40);(1,2,5,4)(3,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,23)(22,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,3)(2,6)(4,8)(5,7)(9,11,12,15)(10,14,13,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,33)(28,37)(29,30)(31,35)(32,36)(34,38)(39,40);(1,4,5,2)(3,8,7,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|b^2=h,c^2=ghi,d^2=fgh,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, 3173233);