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 | |
4534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10,14,15)(11,18,16,12)(13,17,22,23)(19,24,21,20)(25,26,31,30)(27,32,34,29)(28,33,37,38)(35,39,40,36);(3,7)(6,8)(10,15)(11,16)(17,23)(19,21)(25,27,31,34)(26,32,30,29)(28,35,37,40)(33,39,38,36);(1,2,5,4)(3,6,7,8)(10,15)(12,18)(17,23)(20,24)(27,34)(29,32)(35,40)(36,39);(1,3,5,7)(2,6,4,8)(9,11,14,16)(10,12,15,18)(13,21,22,19)(17,24,23,20)(25,28)(26,33)(27,35)(29,36)(30,38)(31,37)(32,39)(34,40);(1,4,5,2)(3,8,7,6)(9,12)(10,16)(11,15)(13,20)(14,18)(17,21)(19,23)(22,24)(25,29)(26,27)(28,36)(30,34)(31,32)(33,35)(37,39)(38,40);(9,13,14,22)(10,17,15,23)(11,19,16,21)(12,20,18,24)(25,30,31,26)(27,29,34,32)(28,33,37,38)(35,39,40,36)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gh,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8458170);