Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512565 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×AΣL(1,𝔽16) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| prime radical | #60388 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
1498 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 1498 conjugacy classes of subgroups — too many to draw. Only its 192 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 — 4 generators on 32 points:
(1,2)(3,12,14,7)(4,8)(5,10,19,9)(6,11)(13,26,28,20)(15,25,18,21)(16,22)(17,24,29,23)(27,32,30,31);(1,3)(2,7)(4,13)(5,18)(6,15)(8,20)(9,12)(10,25)(11,21)(14,19)(16,27)(17,30)(22,31)(23,26)(24,32)(28,29);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,23)(10,24)(11,22)(12,26)(14,28)(15,27)(18,30)(19,29)(21,31)(25,32);(4,16)(8,22)(13,27)(17,29)(20,31)(23,24)(26,32)(28,30)Polycyclic presentation:
2^9|a^2=f,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,f^b=fgi,g^b=gi,h^b=hi,d^c=di,f^e=fhi,g^e=gi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72317553, #72317569, #72317627, … |
| maximal normal subgroup of | #72317553, #72317569, #72317627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419761);