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 | |
4694 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)(11,19)(12,16)(13,17)(14,18)(15,20)(21,22)(23,24)(25,26,29,33)(27,31,35,30)(28,32,37,40)(34,39,36,38);(1,2,4,5)(3,6,7,8)(9,11,14,20)(10,15,18,19)(12,13,22,23)(16,24,21,17)(25,27)(26,30)(28,36)(29,35)(31,33)(32,39)(34,37)(38,40);(11,20)(13,23)(15,19)(17,24)(25,27,29,35)(26,30,33,31)(28,36,37,34)(32,39,40,38);(9,12)(10,16)(11,13)(14,22)(15,24)(17,19)(18,21)(20,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,37)(38,40);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,21)(13,17)(14,18)(16,22)(19,20)(23,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,39)(37,40);(9,13)(10,17)(11,12)(14,23)(15,21)(16,19)(18,24)(20,22)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,40)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=h,e^2=gi,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, 7733867);