Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #236683 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(3,9)(4,11)(6,14)(8,12)(10,13)(15,16)(17,18)(19,28)(20,33)(21,38)(22,26)(23,27)(24,30)(25,36)(29,46)(31,41)(32,40)(34,45)(35,44)(37,43)(39,47)(42,48);(1,2)(3,6)(4,12)(5,7)(8,11)(9,16)(10,15)(13,14)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,42)(26,40)(27,41)(28,38)(29,35)(33,39)(36,47)(37,48)(43,46)(44,45);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,11)(12,14)(17,20)(18,25)(19,29)(21,35)(22,36)(23,37)(24,39)(26,43)(27,44)(28,45)(30,34)(31,48)(32,47)(33,38)(40,42)(41,46);(2,7)(3,10)(8,12)(9,13)Polycyclic presentation:
2^9|e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,d^c=di,e^c=ei,f^c=fhi,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 143879);