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 | #42429 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4399 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)(11,15)(12,20)(13,17)(14,18)(16,21)(19,22)(23,24)(25,26)(27,34)(28,36)(29,31)(30,32)(33,38)(35,39)(37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,30,29)(26,31,32,34)(28,37,38,35)(33,40,36,39);(11,18)(14,15)(19,24)(22,23)(26,32)(31,34)(33,36)(39,40);(9,10)(11,15)(12,16)(13,17)(14,18)(19,23)(20,21)(22,24)(25,28,30,38)(26,33,32,36)(27,35,29,37)(31,39,34,40);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,38)(37,40);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,17,20)(11,19,18,24)(14,22,15,23)(25,29,30,27)(26,34,32,31)(28,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7551350);