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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,11)(15,17)(16,18)(22,23)(25,26)(27,28)(29,37)(30,31)(32,38)(33,34)(35,40)(36,39);(1,2)(3,5)(4,6)(7,8)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,27)(26,28)(29,35)(30,33)(31,34)(32,36)(37,40)(38,39);(1,2,4,6)(3,7,8,5)(12,19)(13,21)(15,17)(16,18)(26,31)(27,33)(29,38)(36,40);(3,8)(5,7)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23)(25,28)(26,27)(29,36)(30,34)(31,33)(32,35)(37,39)(38,40);(1,3)(2,5)(4,8)(6,7)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,26)(27,34)(28,33)(29,32)(30,31)(35,40)(36,39)(37,38);(1,2)(3,5)(4,6)(7,8)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,37)(33,39)(34,40)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8849999);