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 | |
4632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10,14,17)(11,18,12,15)(13,16,21,23)(19,24,20,22);(1,2,5,8)(3,6,4,7)(9,11,14,12)(10,15,17,18)(13,19,21,20)(16,22,23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(1,3,5,4)(2,6,8,7)(13,21)(16,23)(19,20)(22,24)(26,31)(29,39)(32,37)(33,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,18)(13,16)(14,17)(19,22)(20,24)(21,23)(25,27)(26,32)(28,34)(29,33)(30,36)(31,37)(35,38)(39,40);(1,4,5,3)(2,7,8,6)(9,12,14,11)(10,15,17,18)(13,20,21,19)(16,22,23,24)(25,28)(26,33)(27,35)(29,37)(30,38)(31,40)(32,39)(34,36);(9,13,14,21)(10,16,17,23)(11,19,12,20)(15,22,18,24)(25,29,30,39)(26,34,31,35)(27,33,36,40)(28,37,38,32)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8229566);