Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132549 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C2×C4×Q8 | C23 |
| prime radical | #53861 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
576 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 576 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,13,18,14,19,27,35)(11,15,21,31,22,32,26,38)(12,17,25,34,24,16,28,33)(20,30,36,40,23,29,37,39);(1,2)(3,7)(4,6)(5,8)(9,11,14,22)(10,15,19,32)(12,23,24,20)(13,26,27,21)(16,29,17,30)(18,31,35,38)(25,36,28,37)(33,39,34,40);(3,8)(5,7)(10,16)(12,24)(15,29)(17,19)(18,34)(20,23)(25,28)(30,32)(31,40)(33,35)(36,37)(38,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,e^c=ei,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, 39751);