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 | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34,38,39)(35,36,42,40)(37,41,46,47)(43,44,45,48);(9,11,13,16)(10,14,12,15)(19,26)(20,29)(22,25)(23,28)(34,39)(35,42)(41,47)(43,45);(10,12)(14,15)(17,19,21,26)(18,22,24,25)(20,27,29,32)(23,30,28,31)(33,35,38,42)(34,40,39,36)(37,45,46,43)(41,44,47,48);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,15,16,14)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,36,38,40)(34,35,39,42)(37,44,46,48)(41,43,47,45);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,37)(34,41)(35,43)(36,44)(38,46)(39,47)(40,48)(42,45)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,d^2=fgh,e^2=fghi,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, 5479995);