Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 | |
4548 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)(3,5)(4,6)(7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,22)(20,24)(21,23)(25,26)(27,34)(28,38)(29,32)(30,33)(31,37)(35,39)(36,40);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,34)(38,40);(1,3,4,8)(2,5,6,7)(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,19,24)(25,28,30,29)(26,32,33,38)(27,35,37,36)(31,39,34,40);(9,12,14,13)(10,17,18,16)(11,20,22,21)(15,24,19,23)(25,28,30,29)(26,32,33,38)(27,36,37,35)(31,40,34,39);(2,6)(5,7)(9,12,14,13)(10,17,18,16)(11,21,22,20)(15,23,19,24)(25,29,30,28)(26,32,33,38)(27,35,37,36)(31,40,34,39);(9,13,14,12)(10,17,18,16)(11,21,22,20)(15,24,19,23)Polycyclic presentation:
2^9|c^2=gi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013102);