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 | #35071 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4295 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)(9,10,13,15)(11,16,18,14)(12,20,21,17)(19,22,24,23)(29,37)(32,36)(34,40)(38,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,16)(17,22)(20,23)(21,24)(25,26)(27,31)(28,30)(29,32)(33,35)(34,38)(36,37)(39,40);(10,15)(11,18)(17,20)(19,24)(26,28)(31,33)(32,36)(38,39);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,19)(13,18)(14,15)(17,23)(20,22)(21,24)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(9,10)(11,16)(12,17)(13,15)(14,18)(19,23)(20,21)(22,24)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,17,15,20)(11,19,18,24)(14,22,16,23)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=h,d^2=gi,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7626482);