Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139532 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | #2021 | C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10,14,18)(11,19,23,36)(12,24,20,17)(13,26)(15,30,32,21)(16,29)(22,38)(25,34)(27,35)(28,33)(31,39)(37,40);(2,6,8,7)(9,11,20,30,14,23,12,21)(10,15,24,36,18,32,17,19)(13,27,28,37,29,22,25,39)(16,31,33,38,26,40,34,35);(10,16)(15,31)(17,34)(18,26)(19,35)(24,33)(32,40)(36,38)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ehi,d^c=dh,f^c=fh,e^d=ei,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72284469, #72284490, #72284501, … |
| maximal normal subgroup of | #72284469, #72284490, #72284501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46822);