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 | #42997 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4671 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)(11,12)(13,18)(14,16)(15,17)(19,20)(21,23)(22,24)(25,26,31,35)(27,36,37,32)(28,29,33,39)(30,34,40,38);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,16)(13,21)(14,17)(18,20)(19,22)(23,24)(25,27,31,37)(26,32,35,36)(28,30,33,40)(29,38,39,34);(10,16)(11,17)(13,22)(20,23)(27,37)(30,40)(32,36)(34,38);(9,11)(10,12)(13,19)(14,17)(15,16)(18,20)(21,22)(23,24)(25,28)(26,29)(27,30)(31,33)(32,38)(34,36)(35,39)(37,40);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,16)(13,20)(14,15)(18,21)(19,24)(22,23)(25,29)(26,33)(27,38)(28,35)(30,36)(31,39)(32,40)(34,37);(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,24)(17,21)(25,30)(26,34)(27,28)(29,36)(31,40)(32,39)(33,37)(35,38)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,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, 7733684);