Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(6,10)(7,13)(12,17)(15,19)(21,22,25,26)(23,27,31,29)(24,32,34,28)(30,36,33,35)(37,38,42,45)(39,40,47,43)(41,44,50,52)(46,48,49,51);(8,16)(12,17)(15,19)(18,20)(22,26)(27,29)(28,32)(35,36)(37,39,42,47)(38,43,45,40)(41,49,50,46)(44,48,52,51);(5,6)(7,14)(8,12)(9,10)(11,13)(15,20)(16,17)(18,19)(21,23,25,31)(22,27,26,29)(24,33,34,30)(28,36,32,35)(37,39)(38,43)(40,45)(41,49)(42,47)(44,48)(46,50)(51,52);(1,2,3,4)(5,7)(6,11)(8,15)(9,13)(10,14)(12,18)(16,19)(17,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,34)(33,35)(37,40,42,43)(38,39,45,47)(41,48,50,51)(44,46,52,49);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,24,25,34)(22,28,26,32)(23,30,31,33)(27,35,29,36)(37,41)(38,44)(39,46)(40,48)(42,50)(43,51)(45,52)(47,49)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fh,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5188216);