Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2202 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 756 conjugacy classes of subgroups — too many to draw. Only its 162 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:
(1,2,4,3)(9,10)(11,17)(12,15)(13,21)(14,18)(16,20)(19,24)(22,23);(5,6,7,8)(10,15,18,16)(11,13)(17,23,21,24)(19,22);(1,3,4,2)(9,11)(12,19)(13,14)(15,16)(17,21)(20,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72271518, #72271527, #72271531, … |
| maximal normal subgroup of | #72271518, #72271527, #72271531, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54540);