Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105681 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
884 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,23,28,39)(20,30)(21,33)(22,25)(24,32)(26,36)(27,43,45,47)(29,41,34,38)(31,42)(35,40)(37,46,48,44);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23,25,39)(20,27)(21,34)(22,28)(24,37,40,48)(26,41,33,38)(29,36)(30,47,42,43)(31,45)(32,46)(35,44);(3,9)(6,12)(10,15)(13,16)(17,20)(18,24)(19,27)(21,35)(22,31)(23,37)(25,40)(26,42)(28,45)(29,46)(30,33)(32,36)(34,44)(38,47)(39,48)(41,43)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ci,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12877);