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 | #42679 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4384 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)(11,18)(12,13)(14,16)(15,19)(17,22)(20,21)(23,24)(26,31)(27,34)(33,37)(35,40);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,20)(13,23)(14,19)(16,18)(17,21)(22,24)(25,26,30,31)(27,32,34,28)(29,37,39,33)(35,36,40,38);(1,3,5,4)(2,7,6,8)(10,16)(11,19)(12,22)(21,23)(26,31)(28,32)(33,37)(36,38);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,20)(13,16)(14,22)(15,21)(18,23)(19,24)(25,27)(26,32)(28,31)(29,35)(30,34)(33,38)(36,37)(39,40);(9,11)(10,15)(12,20)(13,23)(14,19)(16,18)(17,21)(22,24)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,40,37,35);(9,13,14,17)(10,12,16,22)(11,21,19,23)(15,24,18,20)(25,29,30,39)(26,33,31,37)(27,35,34,40)(28,36,32,38)Polycyclic presentation:
2^9|b^2=gh,c^2=gi,d^2=gi,e^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7896814);