Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D16 |
| commutator | C16 | C2×C16 |
| Frattini, cosocle | C8×C16 | C22 |
| socle, prime radical | C22 | #1331 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C8 ◁ C82 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23)(19,27)(20,29)(21,33)(24,39)(25,42)(26,38)(28,32)(30,46)(31,36)(34,40)(35,37)(41,47)(43,44)(45,48);(17,18,22,26)(19,28,31,45)(20,32,34,48)(21,35,33,25)(23,36,38,27)(24,40,41,29)(30,47,43,39)(37,44,42,46)Polycyclic presentation:
2^9|a^2=d,b^2=i,c^2=eg,d^2=f,e^2=gi,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ce,e^b=eg,g^b=gi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 944);