Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #127265 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18)(19,29)(20,34)(21,25)(22,39)(23,28)(24,31)(26,37)(27,36)(30,43)(32,42)(33,48)(35,40)(38,44)(41,47)(45,46);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,21,32,22,33,37,31,23,30,38,46,36,47,20,35)(18,24,26,41,27,42,25,43,28,40,34,48,39,45,44,29);(19,30)(24,40)(29,43)(31,35)(32,46)(33,47)(41,48)(42,45)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=ghi,f^2=g,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=ci,d^b=di,e^b=eh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240741, #72240742, #72240745, … |
| maximal normal subgroup of | #72240741, #72240742, #72240745, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34461);