Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1396 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,12,15,16)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(34,38)(36,44)(39,41)(42,48);(3,7)(5,8)(11,15)(12,16)(18,19)(20,27)(24,26)(28,32)(33,34,37,38)(35,41,43,39)(36,45,44,40)(42,46,48,47);(1,3,4,7)(2,5,6,8)(9,11)(10,12)(13,15)(14,16)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,34)(35,39)(36,45)(37,38)(40,44)(41,43)(42,47)(46,48);(9,12)(10,11)(13,16)(14,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,2)(3,5)(4,6)(7,8)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,36)(34,40)(35,42)(37,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6144542);