Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | #1790 |
| prime radical | C2×#127 | C2×C4 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
618 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 618 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,23)(20,24,28,22)(21,25)(26,31,29,30)(27,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,22)(20,28)(23,30)(24,25)(26,29)(31,32);(3,9)(6,12)(10,15)(13,16)(17,19,21,27)(18,23,25,32)(20,29,28,26)(22,30,24,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=di,e^b=egi,f^b=fgi,d^c=dg,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36838);