Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #427 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8283 |
| prime radical | C23 | #443 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×M32 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 120 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:
(2,7,9,21,11,22,10,17)(8,20,23,28,12,25,24,31);(1,2)(3,12,15,8)(4,17)(5,9)(6,11)(7,13)(10,18)(14,28,30,31)(16,24,29,23)(19,21)(20,26,25,32)(22,27);(1,3,13,26,5,16,19,30,6,15,27,32,18,29,4,14)(2,8,7,20,9,23,21,28,11,12,22,25,10,24,17,31)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=h,g^2=i,b^a=bd,c^b=ch,d^b=dg,e^b=eg,g^b=gi,h^b=hi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72214092, #72214178, #72214318, … |
| maximal normal subgroup of | #72214092, #72214178, #72214318, … |
| maximal quotient of | #72216215, C3×#129159, #72225834, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36355);