Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132201 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | C22×D4 | C22⋊C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#557 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
602 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 602 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(2,6,8,11)(7,15,16,14)(17,18,21,26,22,27,34,42)(19,23,30,37,28,38,33,46)(20,25,32,41,31,24,35,40)(29,36,44,47,43,39,45,48);(1,2)(3,7)(4,11)(5,8)(6,13)(9,14)(10,16)(12,15)(17,19)(18,23)(20,29)(21,33)(22,28)(24,39)(25,36)(26,37)(27,38)(30,34)(31,43)(32,45)(35,44)(40,48)(41,47)(42,46);(1,3,5,10)(2,7,8,16)(4,12,13,9)(6,14,11,15)(18,24)(19,28)(20,31)(23,36)(25,27)(26,41)(30,33)(32,35)(37,48)(38,39)(40,42)(46,47)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^c=fgi,f^d=fi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39565);