Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130482 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | #2397 | C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#556 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
681 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 681 conjugacy classes of subgroups — too many to draw. Only its 82 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:
(2,6,7,8)(9,10,13,18,14,15,28,31)(11,19,21,37,22,34,27,39)(12,23,25,36,26,17,29,35)(16,32,33,24,30,38,40,20);(1,2)(3,6)(4,8)(5,7)(10,15)(16,30)(17,23)(19,34);(1,3,5,4)(2,6,7,8)(9,11,14,22)(10,16,15,30)(12,24,26,20)(13,27,28,21)(17,34,23,19)(18,33,31,40)(25,32,29,38)(35,39,36,37)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,f^b=fgh,e^c=ei,f^c=fgi,f^d=fi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72292959, #72292967, #72293121, … |
| maximal normal subgroup of | #72292959, #72292967, #72293121, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 37743);