Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6473167 |
| 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 | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1634 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)(4,11)(5,7)(8,12)(9,13)(10,16)(14,15)(17,18,22,25,23,27,34,26)(19,28,31,39,32,38,30,21)(20,33,36,24,37,40,47,41)(29,43,45,48,46,35,44,42);(1,3)(2,6)(4,10)(5,9)(7,13)(8,14)(11,16)(12,15)(17,19)(18,21)(20,29)(22,30)(23,32)(24,35)(25,38)(26,28)(27,39)(31,34)(33,42)(36,44)(37,46)(40,48)(41,43)(45,47);(2,7)(3,9)(8,11)(10,15)(17,20,34,47,23,37,22,36)(18,24,25,40,27,41,26,33)(19,29,31,45,32,46,30,44)(21,35,38,48,39,43,28,42);(3,9)(6,13)(10,15)(14,16)(17,21)(18,19)(20,35)(22,28)(23,39)(24,29)(25,31)(26,30)(27,32)(33,44)(34,38)(36,42)(37,43)(40,45)(41,46)(47,48);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,14,13,16)(17,22,23,34)(18,25,27,26)(19,30,32,31)(20,36,37,47)(21,38,39,28)(24,40,41,33)(29,44,46,45)(35,48,43,42)Polycyclic presentation:
2^9|a^2=f,c^2=ghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405883);