Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
522 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 522 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(1,2)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,16,14)(17,18)(19,28,32,40)(20,33,37,43)(21,26,38,25)(22,27)(23,30)(24,36)(29,42)(31,41)(34,39)(35,44)(45,47,48,46);(2,6)(4,13)(7,14)(8,9)(10,15)(11,16)(17,19,22,32)(18,23,27,41)(20,34,37,29)(21,31,38,30)(24,42,44,39)(25,40,26,28)(33,48,43,45)(35,46,36,47);(1,3,5,12)(2,7,9,10)(4,11,13,16)(6,14,8,15)(17,20,22,37)(18,24,27,44)(19,29,32,34)(21,36,38,35)(23,39,41,42)(25,43,26,33)(28,45,40,48)(30,47,31,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=eghi,f^b=fgh,d^c=dgh,f^c=fgi,e^d=ei,f^d=fi,f^e=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39799);