Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432881 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2899 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)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,18,22,26)(19,27,28,40)(20,31,32,24)(21,34,36,25)(23,35,37,30)(29,43,42,41)(33,39,47,46)(38,44,45,48);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(17,19)(18,23)(20,29)(21,35)(22,28)(24,38)(25,40)(26,37)(27,34)(30,36)(31,45)(32,42)(33,48)(39,43)(41,46)(44,47);(3,9)(6,12)(10,15)(13,16)(19,28)(21,36)(23,37)(25,34)(29,42)(33,47)(38,45)(39,46);(2,7)(6,12)(8,14)(13,16)(17,20,22,32)(18,24,26,31)(19,29,28,42)(21,33,36,47)(23,38,37,45)(25,39,34,46)(27,41,40,43)(30,44,35,48);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,13,12,16)(20,32)(24,31)(29,42)(33,47)(38,45)(39,46)(41,43)(44,48)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343823);