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 | #47028 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,22)(23,24)(26,31)(28,32)(33,39)(36,37);(1,3,7,6)(2,4,8,5)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(25,26,30,31)(27,32,35,28)(29,33,38,39)(34,37,40,36);(1,4,7,5)(2,6,8,3)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,23)(21,22)(25,27,30,35)(26,32,31,28)(29,34,38,40)(33,37,39,36);(1,4,7,5)(2,3,8,6)(9,10)(11,12)(13,21)(14,17)(15,18)(16,22)(19,24)(20,23)(25,28)(26,27)(29,37)(30,32)(31,35)(33,40)(34,39)(36,38);(1,5,7,4)(2,3,8,6)(9,12)(10,15)(11,17)(13,20)(14,18)(16,23)(19,21)(22,24)(25,26)(27,32)(28,35)(29,33)(30,31)(34,37)(36,40)(38,39);(1,6,7,3)(2,4,8,5)(9,13)(10,16)(11,19)(12,20)(14,22)(15,23)(17,21)(18,24)(25,29)(26,33)(27,34)(28,36)(30,38)(31,39)(32,37)(35,40)Polycyclic presentation:
2^9|b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10318937);