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 | |
4560 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)(9,10,14,15)(11,16,17,12)(13,18,22,23)(19,24,21,20)(25,26)(27,35)(28,32)(29,33)(30,34)(31,36)(37,40)(38,39);(1,2,5,6)(3,7,4,8)(9,11)(10,12)(13,21)(14,17)(15,16)(18,24)(19,22)(20,23)(25,27,30,36)(26,31,34,35)(28,37,38,29)(32,33,39,40);(10,15)(11,17)(13,22)(20,24)(27,36)(29,37)(31,35)(33,40);(9,11)(10,16)(12,15)(13,19)(14,17)(18,24)(20,23)(21,22)(25,26)(27,31)(28,39)(29,33)(30,34)(32,38)(35,36)(37,40);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,17,15,11)(13,20,22,24)(18,21,23,19)(25,28,30,38)(26,32,34,39)(27,37,36,29)(31,33,35,40);(1,4,5,3)(2,8,6,7)(9,13,14,22)(10,18,15,23)(11,19,17,21)(12,20,16,24)(25,29,30,37)(26,33,34,40)(27,38,36,28)(31,32,35,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7719291);