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 | #2541 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4373 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(12,20)(16,23)(19,24)(21,22)(26,29)(28,32)(33,37)(36,38);(3,7)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,31,29)(27,32,35,28)(30,33,39,37)(34,38,40,36);(1,2,5,4)(3,6,7,8)(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,24)(22,23)(25,27,31,35)(26,32,29,28)(30,34,39,40)(33,38,37,36);(1,3,5,7)(2,6,4,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,28)(26,27)(29,35)(30,38)(31,32)(33,40)(34,37)(36,39);(1,4,5,2)(3,8,7,6)(9,12,14,20)(10,16,18,23)(11,19,17,24)(13,21,15,22)(25,29,31,26)(27,28,35,32)(30,37,39,33)(34,36,40,38);(9,13,14,15)(10,17,18,11)(12,21,20,22)(16,24,23,19)(25,30,31,39)(26,33,29,37)(27,34,35,40)(28,36,32,38)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=h,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366351);