Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #140220 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1926 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #460 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
250 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,7)(3,8)(4,5)(10,14,15,22)(11,17)(12,20)(16,23,21,19)(18,24);(1,2,4,7,6,8,5,3)(11,18)(14,22)(17,24)(19,23);(1,3,5,8,6,7,4,2)(9,10,12,21,13,15,20,16)(11,19,17,14,18,23,24,22)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=dh,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,e^b=ei,d^c=di,e^c=egh,f^c=fi,h^c=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72309600, #72309638, #72309643, … |
| maximal normal subgroup of | #72309600, #72309638, #72309643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47416);