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 | #37559 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4402 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)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(1,3,5,8)(2,6,4,7)(26,30)(27,34)(32,36)(35,40);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,22)(18,23)(25,26,29,30)(27,31,34,33)(28,36,37,32)(35,39,40,38);(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,28)(26,32)(27,35)(29,37)(30,36)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=g,d^2=hi,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7670637);