Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1464 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)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)(33,34,38,39)(35,36,40,43)(37,45,47,41)(42,46,48,44);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(18,19)(23,25)(24,26)(31,32)(34,39)(36,43)(41,45)(44,46);(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,35)(34,36)(37,42)(38,40)(39,43)(41,44)(45,46)(47,48);(1,3)(2,6)(4,8)(5,7)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,36)(34,40)(35,39)(37,46)(38,43)(41,42)(44,47)(45,48);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,37,38,47)(34,41,39,45)(35,42,40,48)(36,44,43,46)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,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, 2705242);