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 | #49582 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4489 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,16)(12,18)(13,21)(14,15)(17,22)(19,20)(23,24)(25,26)(27,31)(28,29)(30,32)(33,34)(35,36)(37,40)(38,39);(3,8)(5,6)(10,15)(11,18)(17,21)(19,24)(25,27)(26,31)(28,39)(29,38)(30,36)(32,35)(33,40)(34,37);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(13,22)(20,23)(26,32)(27,36)(28,34)(38,40);(1,3,4,8)(2,6,7,5)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,28)(26,33)(27,37)(29,32)(30,34)(31,38)(35,40)(36,39);(1,3)(2,6)(4,8)(5,7)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,29,30,33)(26,34,32,28)(27,38,36,40)(31,37,35,39);(1,2)(3,6)(4,7)(5,8)(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,27)(26,35)(28,37)(29,38)(30,36)(31,32)(33,40)(34,39)Polycyclic presentation:
2^9|d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841893);