Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1461 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(10,12)(14,15)(19,25)(20,27)(22,28)(23,30)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,48)(45,46);(1,2)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14)(19,25)(22,28)(26,31)(29,32)(34,38)(39,41)(40,46)(47,48);(2,4)(6,7)(9,11,13,16)(10,14,12,15)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35,37,43)(34,39,38,41)(36,44,45,42)(40,48,46,47);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,14,16,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,47)(45,46);(1,4,5,2)(3,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,36,37,45)(34,40,38,46)(35,42,43,44)(39,47,41,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fghi,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6159867);