Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6362510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1515 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(17,18,23,28)(19,29,32,43)(20,33,36,25)(21,27,22,26)(24,37,40,31)(30,41,47,45)(34,39,46,48)(35,42,38,44);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19)(18,24)(20,34)(21,31)(22,37)(23,32)(25,41)(26,29)(27,43)(28,40)(30,35)(33,45)(36,46)(38,47)(39,42)(44,48);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11);(17,20)(18,25)(19,30)(21,35)(22,38)(23,36)(24,39)(26,42)(27,44)(28,33)(29,45)(31,46)(32,47)(34,37)(40,48)(41,43);(2,9)(7,13)(8,16)(11,15)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,35,36,38)(24,29,40,43)(25,42,33,44)(30,46,47,34)(39,45,48,41)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ehi,f^a=fh,g^a=gi,d^b=df,f^b=fh,g^c=gi,f^d=fh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6269783);