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 | #43489 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4541 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,14,16)(11,19,20,15)(12,17,18,13)(21,23,24,22)(25,26)(27,37)(28,33)(29,34)(30,35)(31,36)(32,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,21)(13,23)(14,20)(16,19)(17,22)(18,24)(25,27,31,38)(26,32,36,37)(28,30,34,40)(29,35,33,39);(10,16)(11,20)(12,18)(22,23)(27,38)(30,40)(32,37)(35,39);(1,2,5,4)(3,6,8,7)(9,12,14,18)(10,17,16,13)(11,21,20,24)(15,22,19,23)(25,28,31,34)(26,33,36,29)(27,30,38,40)(32,39,37,35);(1,4,5,2)(3,7,8,6)(9,13,14,17)(10,18,16,12)(11,22,20,23)(15,21,19,24)(25,29,31,33)(26,34,36,28)(27,39,38,35)(30,37,40,32);(9,11)(10,19)(12,21)(13,22)(14,20)(15,16)(17,23)(18,24)(25,30)(26,35)(27,34)(28,38)(29,37)(31,40)(32,33)(36,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7652532);