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 | |
1312 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)(3,5)(4,6)(7,8)(11,15)(13,16)(17,18,22,20)(19,25,26,23)(21,28,29,24)(27,30,32,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,46)(45,47);(3,7)(5,8)(9,10,12,14)(11,13,15,16)(19,26)(21,29)(23,25)(24,28)(34,39)(35,41)(37,45)(44,48);(1,2,4,6)(3,5,7,8)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31)(33,35,38,41)(34,40,39,36)(37,46,45,43)(42,44,47,48);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,13,14,16)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31)(33,36,38,40)(34,41,39,35)(37,44,45,48)(42,46,47,43);(1,3)(2,5)(4,7)(6,8)(17,21,22,29)(18,24,20,28)(19,27,26,32)(23,30,25,31)(33,37)(34,42)(35,43)(36,44)(38,45)(39,47)(40,48)(41,46)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=fg,d^2=hi,e^2=fh,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, 5429424);