Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129125 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1776 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8276 |
| prime radical | C23 | #430 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
223 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 223 conjugacy classes of subgroups — too many to draw. Only its 116 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:
(1,2,4,12,7,14,23,13,9,16,5,24,8,15,6,11)(3,17,18,28,20,27,22,29,21,30,19,31,26,32,25,10);(1,3,6,19,8,20,5,25,9,21,23,18,7,26,4,22)(2,10,13,17,15,28,12,27,16,29,11,30,14,31,24,32);(1,4,8,6,9,5,7,23)(2,11,14,12,16,13,15,24)(3,18,26,25,21,19,20,22)(10,27,28,30,29,32,31,17)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=di,f^a=fi,c^b=ch,d^b=dg,e^b=eg,g^b=gi,d^c=di,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72214066, #72214092, #72214592, … |
| maximal normal subgroup of | #72214066, #72214092, #72214592, … |
| maximal quotient of | #72216225, C3×#129125, #72226301, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36321);