Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #354 |
| commutator | C42 | C2×C16 |
| Frattini, cosocle | #1831 | C22 |
| socle | C22 | #1319 |
| prime radical | C23 | #323 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1831 ◁ G | |
| LECS | G ▷ #1831 ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18,20,24,22,26,29,31,23,27,30,32,21,25,19,28);(17,19)(18,24,27,32)(20,22)(21,30)(23,29)(25,28,26,31)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=g,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cgi,d^b=degi,e^b=ei,d^c=dgi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 894);