Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| socle | C22 | #1867 |
| prime radical | C23 | #385 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
318 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 318 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2,5,3,4,6,8,7)(9,10)(11,21)(12,18)(13,26)(14,29)(15,20)(16,22)(17,25)(19,31)(23,39)(24,36)(27,34)(28,35)(30,32)(33,40)(37,38);(1,3,5,6,4,7,8,2)(9,11)(10,16)(12,22)(13,27)(14,30)(15,24)(17,32)(18,36)(19,33)(20,34)(21,26)(23,31)(25,40)(28,38)(29,37)(35,39);(2,6)(3,7)(9,12,15,13)(10,17,20,35)(11,22,24,27)(14,28,31,25)(16,32,34,39)(18,19,26,29)(21,37,36,33)(23,40,30,38)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=egi,d^c=dg,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43159);