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 | |
4696 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,18,13,15)(12,16,20,23)(19,24,21,22)(25,26,30,34)(27,29,33,31)(28,36,38,32)(35,39,40,37);(2,5)(6,7)(9,11,14,13)(10,15,17,18)(12,19,20,21)(16,22,23,24)(25,27,30,33)(26,31,34,29)(28,35,38,40)(32,39,36,37);(1,2)(3,6)(4,5)(7,8)(12,20)(16,23)(19,21)(22,24)(25,28,30,38)(26,32,34,36)(27,35,33,40)(29,37,31,39);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,18)(14,17)(19,22)(20,23)(21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,37)(38,40);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,28)(26,32)(27,35)(29,37)(30,38)(31,39)(33,40)(34,36);(9,13,14,11)(10,15,17,18)(12,21,20,19)(16,22,23,24)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,40,36,35)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^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, 8457000);