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 | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4222 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)(3,7)(4,6)(5,8)(10,11)(15,17)(16,18)(23,24)(25,26,32,36)(27,29,33,28)(30,34,37,39)(31,35,40,38);(1,2,4,6)(3,5,8,7)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(25,27,32,33)(26,28,36,29)(30,31,37,40)(34,38,39,35);(1,2)(3,5)(4,6)(7,8)(25,28,32,29)(26,33,36,27)(30,38,37,35)(31,34,40,39);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,29,32,28)(26,27,36,33)(30,35,37,38)(31,39,40,34);(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,30)(26,34)(27,31)(28,38)(29,35)(32,37)(33,40)(36,39);(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,31)(26,35)(27,37)(28,34)(29,39)(30,33)(32,40)(36,38)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8003630);