Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 |
| prime radical | #36818 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4529 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,22)(13,17)(14,15)(16,19)(20,23)(21,24)(25,26)(27,33)(28,32)(29,31)(30,34)(35,39)(36,38)(37,40);(2,5)(6,8)(10,15)(16,18)(17,22)(23,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,37)(36,40)(38,39);(9,11)(10,16)(12,20)(13,21)(14,19)(15,18)(17,23)(22,24)(26,31)(27,34)(32,38)(35,40);(1,3,4,7)(2,6,5,8)(10,15)(11,19)(17,22)(20,21)(26,31)(30,33)(32,38)(37,39);(9,12,14,13)(10,17,15,22)(11,20,19,21)(16,23,18,24)(25,28,29,36)(26,32,31,38)(27,35,34,40)(30,37,33,39);(9,13,14,12)(10,17,15,22)(11,21,19,20)(16,23,18,24)Polycyclic presentation:
2^9|d^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^c=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7542121);