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 | #43417 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,19)(11,16,20,15)(12,17,22,23)(13,18,24,21)(25,26)(27,33)(28,31)(29,34)(30,32)(35,40)(36,38)(37,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,21)(13,23)(14,20)(16,19)(17,24)(18,22)(25,27,30,28)(26,31,32,33)(29,35,37,36)(34,38,39,40);(11,20)(12,22)(15,16)(17,23)(26,32)(31,33)(34,39)(38,40);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,22)(23,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(1,4,5,3)(2,8,6,7)(9,12,14,22)(10,17,19,23)(11,21,20,18)(13,15,24,16)(25,28,30,27)(26,33,32,31)(29,36,37,35)(34,40,39,38);(9,13)(10,18)(11,17)(12,15)(14,24)(16,22)(19,21)(20,23)(25,29,30,37)(26,34,32,39)(27,35,28,36)(31,38,33,40)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7643830);