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) = C25×C4 |
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 | |
1437 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)(5,8)(9,10,14,12)(11,15,13,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,31,30,32)(33,34)(35,41)(36,43)(37,40)(38,39)(42,46)(44,48)(45,47);(9,11,14,13)(10,15,12,16)(18,19)(21,29)(23,25)(27,30)(34,39)(36,41)(37,45)(42,48);(1,2,4,6)(3,5,7,8)(9,12,14,10)(11,15,13,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,30,32)(33,35,38,43)(34,36,39,41)(37,42,45,48)(40,44,47,46);(1,3,4,7)(2,5,6,8)(9,13,14,11)(10,15,12,16)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31)(33,36)(34,35)(37,46)(38,41)(39,43)(40,48)(42,47)(44,45);(17,21,22,29)(18,24,19,26)(20,27,28,30)(23,31,25,32)(33,37,38,45)(34,40,39,47)(35,42,43,48)(36,44,41,46)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=i,d^2=gi,e^2=f,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, 3077339);