Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470058 |
| 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 | #62814 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2952 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)(17,18,23,28)(19,29,31,42)(20,34,36,40)(21,26,22,27)(24,32,39,33)(25,37,41,35)(30,38,44,48)(43,46,47,45);(1,3)(2,6)(4,11)(5,10)(7,15)(8,14)(9,12)(13,16)(17,19)(18,24)(20,30)(21,32)(22,33)(23,31)(25,38)(26,29)(27,42)(28,39)(34,47)(35,45)(36,44)(37,46)(40,43)(41,48);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,40)(27,34)(28,41)(29,43)(31,44)(32,45)(33,46)(39,48)(42,47);(3,10)(6,14)(9,11)(13,15)(19,31)(24,39)(29,42)(30,44)(32,33)(38,48)(43,47)(45,46);(2,8)(6,14)(7,16)(13,15)(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,36,37)(24,29,39,42)(25,40,41,34)(30,45,44,46)(38,43,48,47)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384624);