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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,32,30,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,47)(46,48);(10,14)(15,16)(18,19)(21,29)(23,25)(27,30)(34,35)(36,43)(37,45)(47,48);(1,2,5,4)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,30,32)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,47,46,48);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31)(33,36,38,43)(34,39,35,41)(37,46,45,44)(40,48,42,47);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,16,14,15)(17,21,22,29)(18,24,19,26)(20,27,28,30)(23,31,25,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,47)(41,48)(43,46)Polycyclic presentation:
2^9|a^2=fg,c^2=fi,d^2=h,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, 3166734);