Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
324 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 112 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)(5,6,9,7,8,10,12,11)(14,20,22,15)(16,17)(18,27)(21,23,25,24)(26,28);(2,4)(5,7,9,10,8,11,12,6)(13,14,18,24,19,22,27,23)(15,16,21,17,20,26,25,28);(13,15,19,20)(14,16,22,26)(17,23,28,24)(18,21,27,25)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=i,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dg,e^b=egi,f^b=fi,g^b=gi,d^c=dgi,e^c=eg,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318512, #72318536, #72318588, … |
| maximal normal subgroup of | #72318512, #72318536, #72318588, … |
| maximal quotient of | #72322446, #72322458 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43105);