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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(9,10,12,15)(11,14,16,13)(17,18,21,24)(19,25,26,22)(20,23,28,31)(27,32,29,30)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(19,26)(20,28)(22,25)(23,31)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(11,16)(13,14)(17,19,21,26)(18,22,24,25)(20,29,28,27)(23,32,31,30)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,21,28)(18,23,24,31)(19,27,26,29)(22,30,25,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,31)(29,32)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fh,d^2=fh,e^2=fgh,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, 5894388);