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 | #43481 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(25,26,30,32)(27,35,37,31)(28,33,38,39)(29,40,36,34);(1,3,5,8)(2,4,7,6)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,23)(22,24)(25,27,30,37)(26,31,32,35)(28,36,38,29)(33,40,39,34);(1,3,5,8)(2,6,7,4)(10,14)(11,18)(12,21)(23,24)(26,32)(28,38)(29,36)(31,35);(2,7)(3,8)(9,11)(10,15)(12,22)(13,18)(14,17)(16,24)(19,21)(20,23)(25,26)(27,35)(28,39)(29,34)(30,32)(31,37)(33,38)(36,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,23)(17,24)(18,22)(25,28)(26,33)(27,36)(29,37)(30,38)(31,40)(32,39)(34,35);(1,4,5,6)(2,3,7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,29,30,36)(26,34,32,40)(27,28,37,38)(31,33,35,39)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8965375);