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 | |
1441 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)(20,25,27,23)(21,26,30,24)(28,31,29,32)(33,34)(35,36)(37,42)(38,39)(40,41)(43,44)(45,48)(46,47);(1,2,5,4)(3,6,7,8)(9,10,13,12)(11,15,16,14)(18,19)(20,27)(24,26)(28,29)(34,39)(35,41)(42,47)(43,48);(9,11,13,16)(10,14,12,15)(17,19,22,18)(20,23,27,25)(21,26,30,24)(28,31,29,32)(33,35)(34,40)(36,39)(37,43)(38,41)(42,45)(44,47)(46,48);(1,3)(2,6)(4,8)(5,7)(9,10,13,12)(11,14,16,15)(17,20,22,27)(18,23,19,25)(21,29,30,28)(24,32,26,31)(33,36,38,40)(34,41,39,35)(37,45,46,44)(42,43,47,48);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,21,22,30)(18,24,19,26)(20,28,27,29)(23,31,25,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,47)(40,45)(41,48)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,c^2=gh,d^2=h,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, 2712138);