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 | |
1550 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,29)(22,23)(24,27)(25,30)(26,32)(28,31)(35,41)(36,44)(38,42)(39,46);(11,16)(13,15)(18,23)(20,27)(21,30)(26,31)(33,34,37,40)(35,42,41,38)(36,39,44,46)(43,48,47,45);(2,3)(6,7)(17,19)(18,20)(21,26)(22,24)(23,27)(25,28)(29,32)(30,31)(33,35,37,41)(34,38,40,42)(36,43,44,47)(39,45,46,48);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,36)(34,39)(35,43)(37,44)(38,45)(40,46)(41,47)(42,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=h,c^2=h,d^2=gh,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, 5908632);