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 | #59841 | C2 |
| 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:
(9,10)(11,13)(12,16)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26)(27,38)(28,35)(29,34)(30,39)(31,33)(32,37)(36,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,18)(10,13,15,17)(12,20,22,19)(16,21,24,23)(25,27,32,31)(26,33,37,38)(28,39,29,36)(30,35,40,34);(2,3)(6,7)(10,15)(11,18)(16,24)(19,20)(25,28,32,29)(26,34,37,35)(27,39,31,36)(30,38,40,33);(4,8)(6,7)(9,12,14,22)(10,16,15,24)(11,19,18,20)(13,23,17,21)(25,29,32,28)(26,35,37,34)(27,39,31,36)(30,38,40,33);(1,3,5,2)(4,7,8,6)(9,11)(10,17)(12,19)(13,15)(14,18)(16,21)(20,22)(23,24)(25,30,32,40)(26,36,37,39)(27,34,31,35)(28,33,29,38);(1,4,5,8)(2,6,3,7)(9,13,14,17)(10,11,15,18)(12,21,22,23)(16,20,24,19)(25,31,32,27)(26,33,37,38)(28,36,29,39)(30,35,40,34)Polycyclic presentation:
2^9|b^2=h,c^2=g,d^2=h,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8558604);