Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #114220 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #60962 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,13)(5,7)(8,14)(10,15,12,16)(17,18)(19,28,32,43)(20,33)(21,26)(22,36)(23,27)(24,30,39,31)(25,34)(29,38,46,40)(35,41)(37,42)(44,47,48,45);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,40)(26,43)(27,39)(28,36)(33,48)(34,47)(35,46)(37,45)(38,42)(41,44);(3,10)(4,14)(6,15)(8,13)(9,16)(11,12)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,41)(27,42)(28,44)(30,45)(31,47)(32,46)(33,36)(39,40)(43,48)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,f^b=fh,g^b=ghi,e^c=ei,f^c=fh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 21416);