Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#106 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
280 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(1,2,4,3)(13,14,16,19,25,26,24,28,18,21,23,27,17,22,15,20);(5,6,7,10,8,9,11,12)(14,19,21,27)(15,23)(16,24)(17,25)(20,22,28,26);(1,3,4,2)(6,9)(10,12)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,f^a=fi,c^b=ch,d^b=dg,e^b=egi,g^b=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72318467, #72318470, #72318478, … |
| maximal normal subgroup of | #72318467, #72318470, #72318478, … |
| maximal quotient of | #72322431 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36301);