Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×D8 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C2 | C2×C8×D8 |
| prime radical | #3414 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×M32 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
481 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 481 conjugacy classes of subgroups — too many to draw. Only its 224 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:
(1,2,6,12,8,13,24,11,9,14,5,23,7,3,15,17)(4,10,20,27,22,28,32,26,18,25,19,31,21,16,29,30);(1,3)(2,8)(4,16)(5,11)(6,17)(7,14)(9,13)(10,22)(12,24)(15,23)(18,28)(19,26)(20,30)(21,25)(27,32)(29,31);(1,4,6,20,8,22,24,32,9,18,5,19,7,21,15,29)(2,10,12,27,13,28,11,26,14,25,23,31,3,16,17,30);(4,18)(10,25)(16,28)(19,20)(21,22)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72214092, #72214100, #72214134, … |
| maximal normal subgroup of | #72214092, #72214100, #72214134, … |
| maximal quotient of | #72216211, #72225391, C3×#483512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390708);