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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(4,8)(6,7)(9,10)(11,15)(12,20)(13,18)(14,16)(17,22)(19,21)(23,24)(25,26,30,34)(27,35,37,31)(28,38,39,32)(29,36,40,33);(1,2,5,3)(4,6,8,7)(9,11,14,18)(10,15,16,13)(12,19,22,24)(17,23,20,21)(25,27)(26,31)(28,33)(29,32)(30,37)(34,35)(36,39)(38,40);(2,3)(6,7)(10,16)(11,18)(17,20)(19,24)(25,28,30,39)(26,32,34,38)(27,33,37,36)(29,35,40,31);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,17)(14,16)(15,18)(19,21)(20,22)(23,24)(25,28)(26,32)(27,36)(29,35)(30,39)(31,40)(33,37)(34,38);(9,12,14,22)(10,17,16,20)(11,19,18,24)(13,21,15,23)(25,29,30,40)(26,33,34,36)(27,32,37,38)(28,35,39,31);(1,4)(2,6)(3,7)(5,8)(9,13)(10,11)(12,21)(14,15)(16,18)(17,19)(20,24)(22,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,37)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=ghi,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8014698);