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 | #38025 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,17)(13,18)(14,19)(16,20)(21,22)(23,24)(26,30)(28,36)(31,33)(34,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,19)(21,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,2,5,4)(3,6,8,7)(11,20)(12,22)(15,16)(17,21)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,18)(13,21)(14,20)(15,19)(17,23)(22,24)(25,27,29,35)(26,31,30,33)(28,34,36,40)(32,38,37,39);(1,4,5,2)(3,7,8,6)(9,12)(10,17)(11,18)(13,15)(14,22)(16,23)(19,21)(20,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39);(9,13,14,23)(10,18,19,24)(11,21,20,17)(12,15,22,16)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7650832);