Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129617 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×#120 | C2×C4 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
561 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 561 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27)(20,31,33,24)(21,34,35,25)(22,26)(23,29,37,44)(28,40,43,47)(30,41)(32,38)(36,42)(39,46)(45,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,30)(18,23,26,37)(20,28,33,43)(21,29,35,44)(24,36,31,45)(25,27,34,41)(32,42,46,48)(38,47,39,40);(3,9)(6,12)(10,15)(13,16)(17,20)(18,24)(19,28)(21,32)(22,33)(23,36)(25,39)(26,31)(27,40)(29,42)(30,43)(34,38)(35,46)(37,45)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ch,e^b=egi,f^b=fg,d^c=dg,f^c=fi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36814);