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 | #34909 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,14,15)(11,17,12,16)(13,21,22,18)(19,23,20,24)(25,26,30,31)(27,28,33,36)(29,32,37,38)(34,35,39,40);(3,7)(5,8)(10,15)(16,17)(18,21)(23,24)(26,31)(27,33)(32,38)(34,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,12)(10,16,15,17)(13,19,22,20)(18,23,21,24)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,10)(11,17)(12,16)(13,18)(14,15)(19,24)(20,23)(21,22)(25,28)(26,27)(29,35)(30,36)(31,33)(32,34)(37,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,11)(10,17,15,16)(13,20,22,19)(18,24,21,23)(25,29,30,37)(26,32,31,38)(27,34,33,39)(28,35,36,40);(9,13,14,22)(10,18,15,21)(11,19,12,20)(16,23,17,24)Polycyclic presentation:
2^9|a^2=hi,c^2=gi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8079927);