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 | #44138 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,14,16)(11,15,20,17)(12,13,19,18)(21,23,24,22)(26,31)(27,33)(34,37)(35,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,21)(13,23)(14,20)(16,17)(18,22)(19,24)(25,26)(27,32)(28,33)(29,37)(30,31)(34,39)(35,36)(38,40);(1,3,5,4)(2,6,8,7)(10,16)(11,20)(12,19)(22,23)(26,31)(28,32)(34,37)(36,38);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,21)(13,22)(14,20)(15,16)(18,23)(19,24)(25,27,30,33)(26,32,31,28)(29,35,39,40)(34,38,37,36);(1,3,5,4)(2,6,8,7)(9,12)(10,18)(11,21)(13,16)(14,19)(15,22)(17,23)(20,24)(25,28,30,32)(26,33,31,27)(29,38,39,36)(34,35,37,40);(1,4,5,3)(2,7,8,6)(9,13,14,18)(10,19,16,12)(11,22,20,23)(15,21,17,24)(25,29,30,39)(26,34,31,37)(27,35,33,40)(28,36,32,38)Polycyclic presentation:
2^9|a^2=gi,c^2=ghi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7882215);