Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12108 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C2×#424 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,4,3)(9,12)(13,14,18,24)(15,26,29,41)(16,20,33,36)(17,25,19,23)(21,34,39,28)(22,32,37,30)(27,40,31,38)(35,43,42,44);(1,3,4,2)(5,6)(7,9)(8,10)(11,12)(14,20)(16,30)(17,19)(21,35)(22,25)(23,37)(24,36)(26,40)(27,43)(28,34)(31,44)(32,33)(38,41)(39,42);(5,7,8,11)(6,9,10,12)(13,15,18,29)(14,21,24,39)(16,31,33,27)(17,34,19,28)(20,35,36,42)(22,40,37,38)(23,41,25,26)(30,44,32,43)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=h,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,e^b=eh,g^b=gi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43239);