Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1738 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1738 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:
(2,4)(9,10)(11,17,18,23)(12,20,21,15)(13,16)(14,19)(22,24);(5,6,7,8)(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(1,2,3,4)(11,18)(14,22)(15,20)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fhi,d^c=dh,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284236, #72284367, #72284453, … |
| maximal normal subgroup of | #72284236, #72284367, #72284453, … |
| maximal quotient of | #72284632, #72284828, #72286383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54685);