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 | #47016 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4263 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)(10,15)(12,22)(16,19)(18,20)(25,26)(27,34)(28,29)(30,33)(31,32)(35,40)(36,37)(38,39);(1,3,5,8)(2,6,4,7)(9,10)(11,19)(12,17)(13,18)(14,15)(16,21)(20,24)(22,23)(26,32)(27,28)(33,39)(35,36);(10,15)(12,22)(16,19)(18,20)(25,27,31,28)(26,29,32,34)(30,36,38,35)(33,40,39,37);(9,11)(10,16)(12,18)(13,23)(14,21)(15,19)(17,24)(20,22)(25,28,31,27)(26,29,32,34)(30,35,38,36)(33,40,39,37);(1,4,5,2)(3,7,8,6)(9,12,14,22)(10,17,15,23)(11,20,21,18)(13,19,24,16)(25,29)(26,27)(28,32)(30,37)(31,34)(33,35)(36,39)(38,40);(9,13)(10,18)(11,17)(12,19)(14,24)(15,20)(16,22)(21,23)(25,30,31,38)(26,33,32,39)(27,35,28,36)(29,37,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=gi,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7995128);