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 | |
4445 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,18)(12,16)(13,22)(14,20)(15,17)(19,23)(21,24)(25,26,30,32)(27,33,35,28)(29,31,37,38)(34,39,40,36);(1,2,5,6)(3,7,8,4)(9,11,15,12)(10,16,17,18)(13,21,23,20)(14,19,24,22)(25,26)(27,33)(28,35)(29,31)(30,32)(34,39)(36,40)(37,38);(2,6)(4,7)(10,17)(14,24)(16,18)(19,22)(29,37)(31,38)(34,40)(36,39);(2,6)(3,8)(9,12,15,11)(10,18,17,16)(13,20,23,21)(14,22,24,19)(25,27,30,35)(26,28,32,33)(29,34,37,40)(31,36,38,39);(1,3)(2,4)(5,8)(6,7)(9,13,15,23)(10,19,17,22)(11,20,12,21)(14,18,24,16)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,39)(38,40);(1,4)(2,3)(5,7)(6,8)(9,14,15,24)(10,20,17,21)(11,19,12,22)(13,18,23,16)(25,29,30,37)(26,31,32,38)(27,34,35,40)(28,36,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295502);