Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,4)(3,7)(11,15)(14,16)(17,18,22,19)(20,25,27,23)(21,24,29,26)(28,31,32,30)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,2,6,4)(3,8,7,5)(10,13)(11,15)(18,19)(20,27)(21,29)(30,31)(36,43)(39,41)(44,48)(46,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,35,38,34)(36,41,43,39)(37,42,45,40)(44,47,48,46);(1,4,6,2)(3,8,7,5)(9,11)(10,14)(12,15)(13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,5,6,8)(2,7,4,3)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=g,c^2=fghi,d^2=fhi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6029310);