Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7333 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #322 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | #1311 | C22 |
| socle, prime radical | C22 | #1313 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 48 points:
(17,18,19,24,20,26,30,40,21,25,31,42,33,37,47,48,22,27,29,41,32,44,46,38,34,43,28,39,45,23,35,36);(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23,25,26,27,37,43,44)(19,28,29,31)(20,32)(24,38,39,48,41,40,42,36)(30,47,46,35)(33,45)Polycyclic presentation:
2^9|a^2=c,b^2=d,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,b^a=be,d^a=dgi,f^a=fi,c^b=cgi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224639, #72224644, #72224647, … |
| maximal normal subgroup of | #72224639, #72224644, #72224647, … |
| maximal quotient of | #72224644, #72225140, C2×#93668, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 864);