Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62752 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #3444 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,11,7,9,5,10)(3,12,13,4,8,15,16,14)(17,18)(19,29)(20,25)(21,28)(22,27)(23,26)(24,31)(30,43)(32,42)(33,41)(34,39)(35,40)(36,38)(37,45)(44,47)(46,48);(1,3,7,8)(2,4,9,14)(5,16,6,13)(10,12,11,15)(17,19,23,32)(18,24,26,38)(20,30,34,46)(21,31,22,36)(25,37,39,47)(27,29,28,42)(33,44,35,45)(40,48,41,43);(1,4)(2,8)(3,9)(5,12)(6,15)(7,14)(10,13)(11,16)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,37)(26,39)(27,41)(28,40)(29,43)(31,45)(32,46)(36,44)(38,47)(42,48);(2,9)(4,14)(10,11)(12,15)(18,26)(24,38)(25,39)(27,28)(29,42)(37,47)(40,41)(43,48);(1,5,7,6)(2,10,9,11)(3,13,8,16)(4,15,14,12)(17,21,23,22)(18,27,26,28)(19,31,32,36)(20,33,34,35)(24,29,38,42)(25,40,39,41)(30,44,46,45)(37,48,47,43)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,d^a=dh,e^a=ehi,f^a=fh,e^b=ei,f^b=fh,d^c=di,e^c=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250476);