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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4237 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)(10,15)(11,16)(13,21)(19,24)(25,26)(27,35)(28,36)(29,33)(30,34)(31,32)(37,40)(38,39);(1,2)(3,7)(4,8)(5,6)(26,32)(27,36)(33,34)(37,38);(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(25,27)(26,28)(29,37)(30,38)(31,36)(32,35)(33,39)(34,40);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,28)(26,27)(29,39)(30,40)(31,35)(32,36)(33,37)(34,38);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,31,30)(26,33,32,34)(27,37,36,38)(28,39,35,40);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)(25,30,31,29)(26,34,32,33)(27,38,36,37)(28,40,35,39)Polycyclic presentation:
2^9|c^2=hi,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7554264);