Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(9,10,13,12)(11,14,16,15)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,12,15)(18,19)(23,25)(24,26)(31,32)(36,45)(39,44)(42,48)(46,47);(9,11)(10,14)(12,15)(13,16)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35,37,43)(34,38,40,41)(36,42,45,48)(39,46,44,47);(1,2)(3,6)(4,7)(5,8)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,45)(46,48);(1,4,5,3)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=ghi,c^2=fg,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4201366);