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 | |
1488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,15)(13,16)(17,18,21,22)(19,25,26,23)(20,24,29,28)(27,31,32,30)(33,34,38,40)(35,41,42,36)(37,45,46,39)(43,44,48,47);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(18,22)(19,26)(24,28)(27,32)(35,42)(36,41)(43,48)(44,47);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,35)(34,36)(37,43)(38,42)(39,44)(40,41)(45,47)(46,48);(9,11)(10,13)(12,15)(14,16)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37,38,46)(34,39,40,45)(35,43,42,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=fghi,d^2=f,e^2=g,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, 4557571);