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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4388 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10,14,15)(11,17,18,16)(12,13,21,22)(19,23,24,20)(25,26)(27,34)(28,31)(29,37)(30,33)(32,38)(35,39)(36,40);(2,6)(3,7)(10,15)(11,18)(13,22)(19,24);(1,3,5,7)(2,4,6,8)(10,15)(12,21)(16,17)(19,24)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,4)(2,7)(3,6)(5,8)(9,11,14,18)(10,16,15,17)(12,19,21,24)(13,20,22,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,12)(10,13)(11,19)(14,21)(15,22)(16,20)(17,23)(18,24)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(9,13)(10,12)(11,20)(14,22)(15,21)(16,19)(17,24)(18,23)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=gi,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10232358);