Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | C2×C82 | C22 |
| socle, prime radical | C23 | D4⋊C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C82 ▷ C42 ▷ C22 ▷ C1 | |
210 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(6,11,12,15)(7,8)(9,17)(13,20,19,14)(16,18)(21,22,24,27,25,30,35,36,26,31,23,32,33,29,34,28);(2,4)(5,6,8,13,17,20,7,15,10,12,18,19,9,14,16,11)(22,27,29,28,31,32,30,36)(23,24)(34,35)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fgh,f^2=h,g^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh,c^b=ch,d^b=dh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216529, #72258136, #72258139, … |
| maximal normal subgroup of | #72216529, #72258136, #72258139, … |
| maximal quotient of | #72216527, #72286530, #72323201 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 138);