Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432876 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,28)(20,25)(21,26)(22,34)(23,27)(24,30)(29,44)(31,38)(32,40)(33,41)(35,39)(36,42)(37,46)(43,47)(45,48);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,19)(18,24)(20,29)(21,30)(22,35)(23,31)(25,37)(26,38)(27,39)(28,34)(32,45)(33,47)(36,46)(40,44)(41,48)(42,43);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,25)(19,29)(21,32)(22,36)(23,33)(24,37)(26,40)(27,41)(28,43)(30,45)(31,47)(34,42)(35,46)(38,44)(39,48);(2,8)(6,14)(7,16)(13,15)(17,21,23,22)(18,26,27,34)(19,30,31,35)(20,32,33,36)(24,38,39,28)(25,40,41,42)(29,45,47,46)(37,44,48,43);(18,27)(24,39)(25,41)(26,34)(28,38)(37,48)(40,42)(43,44)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341902);