Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1560 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)(9,10,13,14)(11,12,15,16)(19,25)(20,27)(22,28)(23,30)(33,34)(35,39)(36,44)(37,38)(40,45)(41,43)(42,47)(46,48);(2,5)(6,8)(9,11,13,15)(10,12,14,16)(19,25)(22,28)(26,31)(29,32)(34,38)(36,45)(39,41)(42,48);(1,2)(3,8)(4,5)(6,7)(9,11)(10,12)(13,15)(14,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,3,4,7)(2,6,5,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=fhi,d^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, 5926676);