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 | C25 | C2×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 | |
865 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 865 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) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2)(3,4)(5,6,8,10,9,12,7,11)(13,14)(15,24,28,36)(16,29,33,39)(17,22,34,21)(18,23)(19,26)(20,32)(25,38)(27,37)(30,35)(31,40)(41,43,44,42);(2,4)(6,10)(7,8)(11,12)(13,15,18,28)(14,19,23,37)(16,30,33,25)(17,27,34,26)(20,38,40,35)(21,36,22,24)(29,44,39,41)(31,42,32,43);(13,16,18,33)(14,20,23,40)(15,25,28,30)(17,32,34,31)(19,35,37,38)(21,39,22,29)(24,41,36,44)(26,43,27,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,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, 38501);