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 | |
4531 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)(9,10)(11,13)(12,20)(14,15)(16,18)(17,21)(19,24)(22,23)(25,26,30,33)(27,28,31,35)(29,32,38,40)(34,36,37,39);(3,7)(6,8)(10,15)(12,21)(13,16)(19,22)(25,27)(26,28)(29,37)(30,31)(32,39)(33,35)(34,38)(36,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,18)(17,20)(19,22)(27,31)(28,35)(34,37)(36,39);(9,11,14,18)(10,16,15,13)(12,22,21,19)(17,23,20,24)(25,28,30,35)(26,31,33,27)(29,36,38,39)(32,37,40,34);(1,3)(2,6)(4,7)(5,8)(9,12,14,21)(10,17,15,20)(11,19,18,22)(13,23,16,24)(25,26)(27,28)(29,32)(30,33)(31,35)(34,36)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,16)(10,18,15,11)(12,23,21,24)(17,22,20,19)(25,29)(26,32)(27,34)(28,36)(30,38)(31,37)(33,40)(35,39)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8746501);