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 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(11,12)(14,15)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,31,30,32)(33,34,37,40)(35,41,42,38)(36,44,45,39)(43,46,48,47);(9,10,13,16)(11,14,12,15)(18,22)(20,29)(23,25)(26,30)(35,42)(36,45)(38,41)(39,44);(9,10)(11,14)(12,15)(13,16)(17,19,21,27)(18,23,22,25)(20,30,29,26)(24,32,28,31)(33,35)(34,38)(36,43)(37,42)(39,46)(40,41)(44,47)(45,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,34)(35,41)(36,39)(37,40)(38,42)(43,47)(44,45)(46,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,20,21,29)(18,24,22,28)(19,26,27,30)(23,31,25,32)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=f,d^2=fgh,e^2=fgi,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, 5261017);