Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 402 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,4)(5,6,8,11,9,7,10,12)(13,14,18,23,20,25,16,27,21,26,17,15,19,24,28,22);(1,2,3,4)(5,7,8,12,9,6,10,11)(13,15,19,22,21,23,20,27)(14,16,25,18,26,28,24,17);(13,14,21,26)(15,16,23,28)(17,22,18,27)(19,25,20,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,d^b=dg,e^b=egi,f^b=fg,g^b=gi,h^b=hi,d^c=dgi,e^c=eg,f^c=fgi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318551, #72318555, #72318588, … |
| maximal normal subgroup of | #72318551, #72318555, #72318588, … |
| maximal quotient of | #72322414, #72322463 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43904);