Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12093 |
| 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 | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | #1790 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
547 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 547 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(2,5,6,7)(9,10)(11,20,24,32)(12,25,29,35)(13,18,30,17)(14,19)(15,22)(16,28)(21,34)(23,33)(26,31)(27,36)(37,39,40,38);(1,2)(3,7)(4,6)(5,8)(9,11,14,24)(10,15,19,33)(12,26,29,21)(13,23,30,22)(16,34,36,31)(17,32,18,20)(25,40,35,37)(27,38,28,39);(9,12,14,29)(10,16,19,36)(11,21,24,26)(13,28,30,27)(15,31,33,34)(17,35,18,25)(20,37,32,40)(22,39,23,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38395);