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 | |
1316 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)(9,10,13,11)(12,15,16,14)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34,38,37)(35,39,41,45)(36,43,44,40)(42,48,47,46);(3,7)(6,8)(9,11,13,10)(12,15,16,14)(19,26)(20,29)(22,25)(23,28)(34,37)(35,41)(40,43)(42,47);(1,2,4,5)(3,6,7,8)(17,19,21,26)(18,22,24,25)(20,27,29,32)(23,30,28,31)(33,35)(34,39)(36,42)(37,45)(38,41)(40,46)(43,48)(44,47);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,25)(20,23)(21,24)(22,26)(27,31)(28,29)(30,32)(33,36,38,44)(34,40,37,43)(35,42,41,47)(39,46,45,48);(1,3)(2,6)(4,7)(5,8)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,37,38,34)(35,39,41,45)(36,43,44,40)(42,46,47,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgi,c^2=g,d^2=f,e^2=h,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, 5396942);