Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,3)(4,8)(11,15)(12,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,38,39)(35,37,41,46)(36,43,45,40)(42,48,47,44);(2,3)(6,7)(9,10,13,14)(11,16,15,12)(19,25)(22,28)(26,31)(29,32)(34,39)(35,41)(40,43)(42,47);(1,2,5,3)(4,6,8,7)(9,11)(10,12)(13,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,35)(34,37)(36,42)(38,41)(39,46)(40,44)(43,48)(45,47);(1,3,5,2)(4,7,8,6)(9,10)(11,12)(13,14)(15,16)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,36,38,45)(34,40,39,43)(35,42,41,47)(37,44,46,48);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,37)(34,41)(35,39)(36,44)(38,46)(40,47)(42,43)(45,48)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,c^2=fi,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4637208);