Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500226 |
| 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 | #62888 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,22,25,21,26,32,34)(19,23,29,36,28,37,33,40)(20,24,31,39,30,38,43,45)(27,35,42,47,41,46,44,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,21,28)(18,23,26,37)(20,27,30,41)(22,33,32,29)(24,35,38,46)(25,40,34,36)(31,44,43,42)(39,48,45,47);(2,7)(3,9)(4,11)(12,16)(17,18)(19,23)(20,24)(21,26)(22,34)(25,32)(27,35)(28,37)(29,40)(30,38)(31,45)(33,36)(39,43)(41,46)(42,48)(44,47);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,12,13,16)(17,20,21,30)(18,24,26,38)(19,27,28,41)(22,31,32,43)(23,35,37,46)(25,39,34,45)(29,42,33,44)(36,47,40,48);(3,9)(6,13)(10,15)(12,16)(20,30)(24,38)(27,41)(31,43)(35,46)(39,45)(42,44)(47,48)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6432090);