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 | |
1371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,7,4)(10,13)(11,15)(17,18,22,23)(19,26,24,20)(21,25,29,30)(27,32,31,28)(34,35)(39,41)(40,42)(46,47);(1,3)(2,6)(4,8)(5,7)(10,13)(14,16)(18,23)(19,24)(21,29)(28,32)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,46,48,47);(1,4,5,6)(2,7,8,3)(9,10)(11,16)(12,13)(14,15)(17,19,22,24)(18,20,23,26)(21,27,29,31)(25,28,30,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,4)(2,3)(5,6)(7,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,23)(21,28)(22,26)(25,31)(27,30)(29,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,3)(2,4)(5,7)(6,8)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=fhi,d^2=fghi,e^2=gh,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, 6038252);