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 | #51932 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,14,17)(11,15,18,22)(12,13,16,21)(19,20,23,24)(25,26,31,35)(27,36,39,32)(28,29,40,33)(30,34,38,37);(1,2)(3,7)(4,8)(5,6)(9,11,14,18)(10,15,17,22)(12,19,16,23)(13,20,21,24)(25,27,31,39)(26,32,35,36)(28,37,40,34)(29,38,33,30);(11,18)(15,22)(19,23)(20,24)(25,28,31,40)(26,33,35,29)(27,37,39,34)(30,36,38,32);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,19)(14,16)(15,20)(17,21)(18,23)(22,24)(25,28)(26,29)(27,37)(30,36)(31,40)(32,38)(33,35)(34,39);(1,4,5,3)(2,8,6,7)(9,13)(10,16)(11,20)(12,17)(14,21)(15,23)(18,24)(19,22)(25,29)(26,28)(27,38)(30,39)(31,33)(32,37)(34,36)(35,40);(9,11)(10,15)(12,19)(13,20)(14,18)(16,23)(17,22)(21,24)(25,30,31,38)(26,34,35,37)(27,29,39,33)(28,36,40,32)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=h,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, 7733465);