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 | #45352 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4539 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(26,32)(28,33)(34,39)(36,40);(3,7)(6,8)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(25,26,31,32)(27,33,29,28)(30,34,38,39)(35,40,37,36);(1,2,5,4)(3,6,7,8)(10,15)(13,17)(16,23)(19,24)(25,27,31,29)(26,33,32,28)(30,37,38,35)(34,36,39,40);(1,3,5,7)(2,6,4,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22)(25,28)(26,27)(29,32)(30,36)(31,33)(34,35)(37,39)(38,40);(1,4,5,2)(3,8,7,6)(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,29,31,27)(26,33,32,28)(30,37,38,35)(34,40,39,36);(9,13,14,17)(10,12,15,22)(11,19,20,24)(16,21,23,18)(25,30,31,38)(26,34,32,39)(27,35,29,37)(28,36,33,40)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8279411);