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 | #38077 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,19)(11,20,13,16)(12,17,22,23)(14,18,24,21)(25,26)(27,30)(28,33)(29,32)(31,36)(34,37)(35,39)(38,40);(1,2,5,6)(3,7,4,8)(9,11,15,13)(10,16,19,20)(12,18,22,21)(14,23,24,17)(26,27)(29,37)(31,33)(35,40);(1,3,5,4)(2,7,6,8)(9,12,15,22)(10,17,19,23)(11,21,13,18)(14,16,24,20)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(9,12)(10,17)(11,18)(13,21)(14,16)(15,22)(19,23)(20,24);(1,4,5,3)(2,8,6,7)(9,13,15,11)(10,16,19,20)(12,21,22,18)(14,23,24,17)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(9,14)(10,18)(11,17)(12,16)(13,23)(15,24)(19,21)(20,22)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gh,c^2=g,e^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7641563);