Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #132072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8456 |
| commutator | #117 | C22×C4 |
| Frattini, cosocle | #405 | C23 |
| prime radical | #62292 | C2 |
| LCS | G ▷ #117 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
576 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 576 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,5,10,15,26,6,8,16,22,18,23,31,32)(3,11,13,28,14,20,30,25,12,27,17,24,29,7,19,21);(2,7)(3,12)(4,15)(5,18)(8,20)(9,24)(10,27)(11,23)(13,19)(16,31)(17,30)(21,32)(22,28)(25,26);(2,8)(3,12)(4,16)(7,20)(10,23)(11,27)(14,29)(15,31)Polycyclic presentation:
2^9|a^2=f,d^2=h,f^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,f^c=fi,e^d=ei,f^d=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72292716, C2×#132072 |
| maximal normal subgroup of | #72292716, C2×#132072 |
| maximal quotient of | C2×#132072, #72292822 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 39268);