Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6356405 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60930 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1474 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,15)(12,16)(17,18,23,27)(19,28,31,38)(20,25,33,42)(21,26,22,35)(24,36,39,30)(29,43,46,48)(32,40,34,41)(37,44,47,45);(1,3,4,11,6,12,5,13)(2,7,14,16,9,15,8,10);(1,2)(3,7)(4,14)(5,8)(6,9)(10,13)(11,16)(12,15)(17,19,23,31)(18,24,27,39)(20,29,33,46)(21,30,22,36)(25,37,42,47)(26,38,35,28)(32,44,34,45)(40,43,41,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,10,15,16)(17,20)(18,25)(19,29)(21,34)(22,32)(23,33)(24,37)(26,41)(27,42)(28,43)(30,45)(31,46)(35,40)(36,44)(38,48)(39,47);(3,12)(7,15)(10,16)(11,13)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,33,34)(24,38,39,28)(25,40,42,41)(29,44,46,45)(37,43,47,48)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=i,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6263648);