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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1376 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)(11,15)(14,16)(17,18)(19,22)(20,25)(21,24)(23,27)(26,29)(28,31)(30,32)(33,34,39,36)(35,42,37,40)(38,41,46,44)(43,48,45,47);(10,13)(14,16)(18,19)(20,27)(24,26)(28,32)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36,39,34)(35,40,37,42)(38,41,46,44)(43,48,45,47);(1,2)(3,6)(4,5)(7,8)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=h,d^2=fg,e^2=fghi,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, 4658775);