Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | #2673 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4361 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,20)(13,17)(14,18)(16,21)(19,24)(22,23)(25,26,30,34)(27,31,35,38)(28,32,29,33)(36,39,37,40);(2,5)(6,8)(9,11,14,13)(10,15,18,17)(12,19,21,22)(16,23,20,24)(25,27)(26,31)(28,37)(29,36)(30,35)(32,40)(33,39)(34,38);(9,12,14,21)(10,16,18,20)(11,19,13,22)(15,23,17,24)(27,35)(28,29)(31,38)(32,33);(9,13,14,11)(10,17,18,15)(12,19,21,22)(16,23,20,24)(25,28,30,29)(26,32,34,33)(27,36,35,37)(31,39,38,40);(10,18)(15,17)(16,20)(23,24)(25,29,30,28)(26,33,34,32)(27,36,35,37)(31,39,38,40);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,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, 7615338);