Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4556 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,17)(13,18)(14,16)(15,20)(21,24)(22,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(1,3,5,4)(2,6,7,8)(9,11,14,20)(10,15,16,19)(12,21,13,22)(17,23,18,24)(25,27,29,34)(26,30,32,38)(28,35,36,33)(31,39,40,37);(1,2)(3,6)(4,8)(5,7)(10,16)(15,19)(17,18)(23,24)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(2,7)(6,8)(10,16)(11,20)(17,18)(21,22)(28,36)(31,40)(33,35)(37,39);(1,4,5,3)(2,8,7,6)(9,12,14,13)(10,17,16,18)(11,21,20,22)(15,23,19,24)(26,32)(30,38)(31,40)(37,39);(9,13,14,12)(10,18,16,17)(11,22,20,21)(15,24,19,23)Polycyclic presentation:
2^9|b^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,d^c=dghi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999904);