Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1442 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(33,34,37,40)(35,38,41,45)(36,43,44,39)(42,48,47,46);(3,7)(6,8)(9,11,12,15)(10,13,14,16)(18,19)(23,25)(24,26)(30,31)(35,41)(38,45)(42,47)(46,48);(2,5)(6,8)(10,14)(13,16)(17,19,22,18)(20,23,28,25)(21,24,29,26)(27,31,32,30)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,48)(44,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,37,44)(34,39,40,43)(35,42,41,47)(38,46,45,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=f,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5394908);