Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(5,8)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(20,28)(30,32)(33,34,37,38)(35,41,43,39)(36,40,44,46)(42,48,45,47);(1,2,4,6)(3,5,7,8)(11,15)(13,16)(18,22)(20,28)(23,26)(27,31)(34,38)(36,44)(39,41)(42,45);(1,2)(3,5)(4,6)(7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48);(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,44)(46,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,c^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5690452);