Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,4,8,7)(9,10)(11,16)(12,15)(13,14)(19,25)(20,27)(22,28)(23,30)(35,41)(39,44)(42,47)(46,48);(1,2,5,6)(3,7,8,4)(10,13)(15,16)(19,25)(22,28)(26,31)(29,32)(34,38)(35,41)(40,45)(42,47);(2,6)(3,8)(9,11,14,12)(10,15,13,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,3)(2,4)(5,8)(6,7)(9,12,14,11)(10,16,13,15)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,4)(2,3)(5,7)(6,8)(9,13,14,10)(11,16,12,15)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,36,37,43)(34,40,38,45)(35,42,41,47)(39,46,44,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=i,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6018130);