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 | |
1369 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)(9,10,13,16)(11,14,15,12)(17,18,22,23)(19,26,24,20)(21,25,29,30)(27,32,31,28)(35,41)(38,42)(43,47)(45,48);(2,5)(6,8)(11,15)(12,14)(18,23)(19,24)(21,29)(28,32)(33,34,37,40)(35,38,41,42)(36,39,44,46)(43,45,47,48);(9,11,13,15)(10,14,16,12)(17,19,22,24)(18,20,23,26)(21,27,29,31)(25,28,30,32)(33,35)(34,38)(36,43)(37,41)(39,45)(40,42)(44,47)(46,48);(1,2)(3,6)(4,5)(7,8)(9,12,13,14)(10,15,16,11)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,30)(29,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,3)(2,6)(4,7)(5,8)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,36,37,44)(34,39,40,46)(35,43,41,47)(38,45,42,48)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=h,d^2=fh,e^2=fg,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, 5876996);