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 | |
1387 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)(3,4)(5,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(3,8)(4,7)(9,11,13,16)(10,14,15,12)(18,19)(20,27)(24,26)(28,32)(34,38)(36,44)(39,41)(42,45);(2,6)(3,8)(17,19,22,18)(20,23,27,25)(21,24,29,26)(28,31,32,30)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,44)(46,47);(1,3,5,8)(2,7,6,4)(9,11)(10,12)(13,16)(14,15)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48);(1,4,5,7)(2,8,6,3)(9,12)(10,11)(13,14)(15,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,36,37,44)(34,40,38,46)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,c^2=h,d^2=fgh,e^2=gh,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, 6121238);