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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1294 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)(3,6)(4,8)(5,7)(11,15)(14,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,27)(28,31)(30,32)(33,34,38,36)(35,41,42,39)(37,44,45,40)(43,46,48,47);(1,3,5,8)(2,6,7,4)(10,13)(11,15)(18,19)(20,27)(21,29)(30,31)(35,42)(37,45)(39,41)(40,44);(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35,38,42)(34,39,36,41)(37,43,45,48)(40,46,44,47);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36,38,34)(35,39,42,41)(37,44,45,40)(43,46,48,47);(1,4)(2,3)(5,6)(7,8)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=g,d^2=fh,e^2=f,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, 6105156);