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 |
| prime radical | #42539 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(10,15)(11,18)(12,20)(21,23)(25,26)(27,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(3,7)(6,8)(9,10,14,15)(11,13,18,16)(12,17,20,22)(19,21,24,23)(25,27,31,29)(26,32,28,35)(30,34,38,37)(33,39,36,40);(1,2,5,4)(3,6,7,8)(9,11,14,18)(10,16,15,13)(12,19,20,24)(17,23,22,21)(25,28,31,26)(27,32,29,35)(30,33,38,36)(34,40,37,39);(1,3,5,7)(2,6,4,8)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,22)(21,24);(1,4,5,2)(3,8,7,6)(9,12,14,20)(10,17,15,22)(11,19,18,24)(13,21,16,23)(25,29,31,27)(26,32,28,35)(30,37,38,34)(33,39,36,40);(9,13,14,16)(10,18,15,11)(12,21,20,23)(17,24,22,19)(25,30)(26,33)(27,34)(28,36)(29,37)(31,38)(32,39)(35,40)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095728);