Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4309 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)(3,7)(4,8)(5,6)(9,10)(11,18)(12,15)(13,21)(14,17)(16,20)(19,23)(22,24)(25,26)(27,33)(28,36)(29,31)(30,34)(32,37)(35,38)(39,40);(2,6)(7,8)(9,11)(10,16)(12,18)(13,22)(14,19)(15,20)(17,24)(21,23)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(1,3,5,4)(2,7,6,8)(9,12,15,10)(11,18,20,16)(13,21,14,17)(19,24,22,23)(26,31)(27,34)(32,36)(35,40);(1,3,5,4)(2,7,6,8)(9,13,15,14)(10,17,12,21)(11,19,20,22)(16,23,18,24)(25,28,29,37)(26,32,31,36)(27,35,34,40)(30,38,33,39);(1,3,5,4)(2,7,6,8)(9,14,15,13)(10,17,12,21)(11,19,20,22)(16,24,18,23);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|c^2=gi,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548687);