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 |
| prime radical | #60807 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1644 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)(17,18,23,26)(19,27,32,28)(20,22,34,37)(21,25,35,40)(24,31,39,30)(29,41,43,42)(33,36,46,47)(38,45,48,44);(1,3)(2,6)(4,10)(5,9)(7,13)(8,14)(11,16)(12,15)(17,19)(18,24)(20,28)(21,29)(22,31)(23,32)(25,38)(26,39)(27,34)(30,37)(33,42)(35,43)(36,45)(40,48)(41,46)(44,47);(2,7)(3,9)(8,11)(10,15)(17,20,22,26,23,34,37,18)(19,28,31,39,32,27,30,24)(21,33,36,40,35,46,47,25)(29,42,45,48,43,41,44,38);(3,9)(6,13)(10,15)(14,16)(17,21)(18,25)(19,29)(20,33)(22,36)(23,35)(24,38)(26,40)(27,41)(28,42)(30,44)(31,45)(32,43)(34,46)(37,47)(39,48);(1,4,5,12)(2,8,7,11)(3,10,9,15)(6,14,13,16)(21,35)(25,40)(29,43)(33,46)(36,47)(38,48)(41,42)(44,45)Polycyclic presentation:
2^9|a^2=i,c^2=gh,e^2=h,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=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, 6405701);