Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 52 points:
(2,4)(6,10)(8,16)(11,13)(14,20)(21,22,26,30)(23,31,32,27)(24,28,35,42)(25,38,39,29)(33,43,36,40)(34,41,46,45)(37,50,48,44)(47,51,49,52);(1,2,3,4)(23,32)(24,35)(27,31)(28,42)(34,46)(37,48)(41,45)(44,50);(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18)(21,23,26,32)(22,27,30,31)(24,36,35,33)(25,34,39,46)(28,43,42,40)(29,41,38,45)(37,49,48,47)(44,52,50,51);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,24,26,35)(22,28,30,42)(23,33,32,36)(25,37,39,48)(27,40,31,43)(29,44,38,50)(34,47,46,49)(41,51,45,52);(5,8)(6,12)(7,14)(9,16)(10,17)(11,18)(13,19)(15,20)(21,22)(23,31)(24,28)(25,29)(26,30)(27,32)(33,43)(34,45)(35,42)(36,40)(37,44)(38,39)(41,46)(47,52)(48,50)(49,51);(21,25,26,39)(22,29,30,38)(23,34,32,46)(24,37,35,48)(27,41,31,45)(28,44,42,50)(33,47,36,49)(40,51,43,52)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622691);