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 | |
4479 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(11,17)(15,18)(19,23)(22,24)(25,26,30,32)(27,33,35,29)(28,36,37,31)(34,38,40,39);(3,7)(6,8)(9,10)(11,18)(12,20)(13,14)(15,17)(16,21)(19,22)(23,24)(25,27,30,35)(26,29,32,33)(28,34,37,40)(31,38,36,39);(1,2)(3,6)(4,5)(7,8)(10,14)(15,18)(16,20)(22,24)(28,37)(31,36)(34,40)(38,39);(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,28,30,37)(26,31,32,36)(27,34,35,40)(29,38,33,39);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,24)(21,23)(25,27)(26,29)(28,34)(30,35)(31,38)(32,33)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,27)(28,38)(30,33)(31,34)(32,35)(36,40)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8659504);