Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6)(7,9)(8,11)(10,12)(13,14,17,23,33,42,31,20,19,25,16,22,18,24,34,38)(15,26,28,44,36,41,32,40,30,37,27,39,29,43,35,21);(1,2,3,4)(14,20)(16,31)(17,34)(18,33)(21,37)(22,24)(23,42)(25,38)(26,40)(27,32)(28,35)(29,36)(39,43)(41,44);(6,9)(7,12)(10,11)(13,15,19,30)(14,21,25,40)(16,32,17,35)(18,36,33,29)(20,37,38,26)(22,41,23,43)(24,44,42,39)(27,34,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,e^2=ghi,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eh,d^b=dg,e^b=eg,f^b=fg,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44735);