Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #38141 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(13,21)(16,20)(18,24)(22,23)(25,26)(27,29)(28,36)(30,34)(31,33)(32,38)(35,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,10,14,12)(11,15,19,17)(13,16,21,20)(18,22,24,23)(25,27,30,33)(26,31,34,29)(28,35,38,40)(32,39,36,37);(10,12)(15,17)(16,20)(22,23)(25,26,30,34)(27,29,33,31)(28,36,38,32)(35,39,40,37);(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,40,36,35)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619020);