Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12093 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1113 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 1113 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,9,12,10,7,8,11)(13,14)(15,22)(16,20,26,19)(17,21)(18,24,28,25)(23,27);(2,4)(5,7,10,6)(8,11,9,12)(13,15)(14,18)(16,25)(17,23)(19,22)(20,27)(21,28)(24,26);(14,19)(15,23)(18,27)(20,21)(22,28)(24,25)Polycyclic presentation:
2^9|a^2=f,b^2=hi,d^2=h,f^2=hi,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, 38652);