Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512636 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#354 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #127 | C24 |
| prime radical | #3498 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
840 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 840 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)(3,12,15,22)(4,8)(5,10,19,9)(6,11)(7,14,21,18)(13,26,28,31)(16,23)(17,25,29,24)(20,27,32,30);(1,3,6,14)(2,7)(4,13,16,27)(5,18,19,15)(8,20)(9,12)(10,21)(11,22)(17,30,29,28)(23,31)(24,26)(25,32);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,24)(10,25)(11,23)(12,26)(14,27)(15,28)(18,30)(19,29)(21,32)(22,31);(4,16)(8,23)(13,27)(17,29)(20,31)(24,25)(26,32)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=h,e^2=hi,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=eh,f^b=fgh,g^b=gi,d^c=di,f^e=fhi,g^f=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72318420, #72318931, #72318941 |
| maximal normal subgroup of | #72318420, #72318931, #72318941 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419832);