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 | #51142 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4270 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)(9,10,15,19)(11,17,13,16)(12,14,18,24)(20,21,22,23)(25,26,31,35)(27,36,28,32)(29,30,34,33)(37,40,39,38);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,19,17)(12,22,18,20)(14,21,24,23)(27,28)(32,36)(37,39)(38,40);(11,13)(16,17)(20,22)(21,23)(25,27,31,28)(26,32,35,36)(29,39,34,37)(30,40,33,38);(9,12)(10,14)(11,20)(13,22)(15,18)(16,23)(17,21)(19,24)(25,28,31,27)(26,32,35,36)(29,39,34,37)(30,38,33,40);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,17,19,16)(12,20,18,22)(14,23,24,21)(25,29,31,34)(26,33,35,30)(27,37,28,39)(32,40,36,38);(9,14)(10,18)(11,21)(12,19)(13,23)(15,24)(16,20)(17,22)(25,30,31,33)(26,34,35,29)(27,38,28,40)(32,39,36,37)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7995173);