Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| 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 | #46289 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4372 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)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,26)(27,31)(28,37)(29,32)(30,33)(34,38)(35,40)(36,39);(2,6)(7,8)(10,14)(15,17)(16,22)(23,24)(25,27)(26,31)(28,35)(29,36)(30,34)(32,39)(33,38)(37,40);(1,3,5,4)(2,7,6,8)(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(27,34)(28,29)(31,38)(32,37);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,28,30,29)(26,32,33,37)(27,35,34,36)(31,39,38,40);(10,14)(15,17)(16,22)(23,24)(25,29,30,28)(26,32,33,37)(27,36,34,35)(31,39,38,40);(1,4,5,3)(2,8,6,7)Polycyclic presentation:
2^9|c^2=g,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7610202);