Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
938 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 938 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(5,8)(10,16)(11,22)(12,13)(14,30)(17,33)(18,20)(19,29)(21,32)(23,38)(25,26)(27,28)(31,35)(34,36)(37,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,17)(12,16)(13,29)(14,19)(15,21)(18,27)(20,30)(22,35)(23,31)(24,33)(25,37)(26,36)(28,32)(34,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,23)(13,24)(14,25)(15,26)(16,31)(18,34)(19,37)(20,35)(21,36)(22,30)(27,39)(28,38)(29,33)(32,40);(9,12,14,27,15,28,30,13)(10,18,20,16,21,29,19,32)(11,23,25,39,26,38,22,24)(17,34,35,31,36,33,37,40)Polycyclic presentation:
2^9|d^2=g,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393516);