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 | C2×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,7,4,8)(9,10,14,15)(11,19,21,16)(12,18,13,17)(20,23,22,24)(25,26)(27,33)(28,36)(29,31)(30,34)(32,37)(35,38)(39,40);(2,6)(7,8)(10,15)(16,19)(17,18)(23,24)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(9,11)(10,16)(12,22)(13,20)(14,21)(15,19)(17,24)(18,23)(26,31)(27,34)(32,36)(35,40);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,15,18)(11,20,21,22)(16,23,19,24)(25,28,29,37)(26,32,31,36)(27,35,34,40)(30,38,33,39);(1,3,5,4)(2,7,6,8)(9,13,14,12)(10,18,15,17)(11,20,21,22)(16,23,19,24);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=gi,f^2=gh,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, 7542030);