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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4488 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)(3,8)(4,6)(5,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(25,26,29,31)(27,33,34,30)(28,36,37,32)(35,38,40,39);(1,3,5,4)(2,6,7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,30)(28,35)(29,34)(31,33)(32,38)(36,39)(37,40);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(26,31)(27,34)(28,37)(38,39);(12,21)(15,20)(18,24)(22,23)(27,34)(30,33)(35,40)(38,39);(10,16)(14,17)(15,20)(22,23)(25,28,29,37)(26,32,31,36)(27,35,34,40)(30,38,33,39);(1,4,5,3)(2,6,7,8)Polycyclic presentation:
2^9|a^2=i,b^2=gi,e^2=i,f^2=gh,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, 7615238);