Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1879 |
| prime radical | C23 | #459 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(2,5,7,8)(9,10,13,19,29,38,27,16,15,21,12,18,14,20,30,34)(11,22,24,40,32,37,28,36,26,33,23,35,25,39,31,17);(1,2,6,5,4,7,3,8)(10,16)(12,27)(13,30)(14,29)(17,33)(18,20)(19,38)(21,34)(22,36)(23,28)(24,31)(25,32)(35,39)(37,40);(9,11,15,26)(10,17,21,36)(12,28,13,31)(14,32,29,25)(16,33,34,22)(18,37,19,39)(20,40,38,35)(23,30,24,27)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,f^c=fh,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44212);