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 | |
4332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,30,35)(27,33,38,31)(28,39,37,32)(29,40,36,34);(1,2,5,6)(3,4,7,8)(11,17)(15,21)(18,23)(22,24)(25,27,30,38)(26,31,35,33)(28,36,37,29)(32,34,39,40);(2,6)(3,7)(10,14)(15,21)(16,19)(22,24)(25,26)(27,31)(28,39)(29,34)(30,35)(32,37)(33,38)(36,40);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,30,37)(26,32,35,39)(27,36,38,29)(31,34,33,40);(1,3)(2,4)(5,7)(6,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,27)(26,33)(28,36)(29,37)(30,38)(31,35)(32,40)(34,39);(1,4)(2,3)(5,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,30,36)(26,34,35,40)(27,37,38,28)(31,32,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9148229);