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 | |
4752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,17)(12,18)(13,16)(14,15)(19,24)(20,21)(22,23)(25,26)(27,34)(28,36)(29,31)(30,33)(32,38)(35,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,18)(10,12,15,17)(13,21,22,19)(16,24,23,20)(25,27)(26,31)(28,37)(29,30)(32,40)(33,34)(35,38)(36,39);(10,15)(11,18)(13,22)(20,24)(27,29)(31,34)(35,37)(39,40);(9,11)(10,12)(13,19)(14,18)(15,17)(16,20)(21,22)(23,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,36)(34,40);(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,23)(22,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,38)(37,39);(1,3,4,7)(2,6,5,8)(9,13,14,22)(10,16,15,23)(11,19,18,21)(12,20,17,24)(25,29,30,27)(26,31,33,34)(28,37,38,35)(32,39,36,40)Polycyclic presentation:
2^9|b^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713911);