Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12090 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C27 |
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 | C2×#2403 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
3185 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 3185 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) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,8,10,9,12,7,11)(13,14)(15,18,20,17)(16,19);(2,4)(6,10)(7,8)(11,12)(14,17)(15,20)(18,19);(14,17)(18,19)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dh,e^b=eg,f^b=fgh,d^c=dgi,f^c=fgi,e^d=ei,f^d=fi,f^e=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72238058, #72267435, #72267757, … |
| maximal normal subgroup of | #72238058, #72267435, #72267757, … |
| maximal quotient of | #72238064, #72338281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38496);