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 | #2615 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4313 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)(25,26)(27,33)(28,38)(29,34)(30,32)(31,35)(36,40)(37,39);(2,6)(3,7)(9,10)(11,16)(12,13)(14,15)(17,22)(18,24)(19,21)(20,23)(25,27,32,35)(26,33,30,31)(28,37,29,36)(34,40,38,39);(1,3,5,7)(2,4,6,8)(10,15)(13,17)(16,23)(19,24)(25,28,32,29)(26,34,30,38)(27,36,35,37)(31,40,33,39);(1,4)(2,7)(3,6)(5,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,23)(17,24)(18,22)(25,29,32,28)(26,34,30,38)(27,37,35,36)(31,40,33,39);(9,12)(10,17)(11,18)(13,15)(14,22)(16,19)(20,21)(23,24)(25,30,32,26)(27,33,35,31)(28,38,29,34)(36,39,37,40);(9,13,14,17)(10,12,15,22)(11,19,20,24)(16,21,23,18)(25,31,32,33)(26,35,30,27)(28,39,29,40)(34,36,38,37)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8279455);