Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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:
(1,2,6,4)(3,8,5,7)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,22)(21,28)(24,30)(25,32)(26,27)(29,31)(33,34,38,37)(35,41,42,39)(36,40,44,45)(43,47,48,46);(1,2,6,4)(3,7,5,8)(9,11,13,10)(12,15,16,14)(18,20)(23,27)(24,28)(31,32)(34,37)(35,42)(36,44)(46,47);(1,3,6,5)(2,7,4,8)(17,19,22,26)(18,23,20,27)(21,29,30,25)(24,32,28,31)(33,35,38,42)(34,39,37,41)(36,43,44,48)(40,46,45,47);(1,4,6,2)(3,7,5,8)(9,12)(10,14)(11,15)(13,16)(17,20,22,18)(19,23,26,27)(21,28,30,24)(25,31,29,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,5,6,3)(2,7,4,8)(17,21,22,30)(18,24,20,28)(19,25,26,29)(23,31,27,32)(33,37,38,34)(35,39,42,41)(36,45,44,40)(43,46,48,47)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgi,c^2=fgh,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5151578);