Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #143938 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #469 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #2065 |
| prime radical | C23 | #465 |
| LCS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
224 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,5,8,4,7,3,6)(9,10)(11,12,31,36)(13,25)(14,20)(15,23)(16,22)(17,24)(18,26,37,27)(19,33,28,32)(21,39)(29,35,30,34)(38,40);(9,11,13,26,15,29,14,28,17,31,39,27,16,30,38,19)(10,18,20,12,22,32,25,34,24,37,40,36,23,33,21,35);(1,2,4,7)(3,8,5,6)(9,12)(10,19)(11,25)(13,37)(14,32)(15,35)(16,34)(17,36)(18,39)(20,29)(21,31)(22,27)(23,26)(24,28)(30,40)(33,38)Polycyclic presentation:
2^9|a^2=deh,b^2=d,c^2=gh,d^2=g,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,g^a=gi,h^a=hi,c^b=cf,e^b=ei,d^c=dg,e^c=eh,f^c=fg,g^c=gi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53158);