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 | |
4325 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)(3,4,8,6)(10,14)(15,21)(16,19)(22,24)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(26,28)(27,34)(32,36)(35,40);(1,3)(2,4)(5,8)(6,7)(11,17)(15,21)(18,23)(22,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,4)(2,3)(5,6)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,39,40,38);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39);(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,d^2=h,e^2=h,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, 8458101);