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 | #47633 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4324 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)(9,10,15,12)(11,17,13,16)(14,18,22,20)(19,24,21,23)(25,26,31,33)(27,35,36,32)(28,38,39,30)(29,40,37,34);(1,2,5,6)(3,7,4,8)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27,31,36)(26,32,33,35)(28,37,39,29)(30,34,38,40);(1,3,5,4)(2,7,6,8)(10,12)(16,17)(18,20)(23,24)(26,33)(27,36)(28,39)(34,40);(9,12,15,10)(11,17,13,16)(14,20,22,18)(19,24,21,23)(25,28)(26,30)(27,37)(29,36)(31,39)(32,34)(33,38)(35,40);(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,29,31,37)(26,34,33,40)(27,28,36,39)(30,32,38,35);(1,4,5,3)(2,8,6,7)(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30)(26,28)(27,34)(29,35)(31,38)(32,37)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7802314);