Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #94871 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C2×#97 |
| LCS | G ▷ C22×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
296 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6,8,7,9,10,12,11)(13,14,17,18,19,21,16,22)(15,23,24,28,26,25,27,20);(2,4)(13,15,18,25,17,27,21,28,19,26,22,23,16,24,14,20);(5,7,8,10,9,11,12,6)(14,21)(15,24,26,27)(18,22)(20,25,28,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=cg,f^a=fi,c^b=ch,e^b=ei,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72318530, #72318532, #72318580, … |
| maximal normal subgroup of | #72318530, #72318532, #72318580, … |
| maximal quotient of | #72322457 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36264);