Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | #1889 |
| prime radical | C2×H(5,𝔽2) | C2×C4 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
591 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 591 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2,4,7,5,3,6,8)(9,10)(11,17,18,24)(12,20,22,15)(13,16)(14,19)(21,23);(1,3,4,8,5,2,6,7)(11,18)(14,23)(15,20)(19,21);(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,e^a=egi,g^a=gi,c^b=ci,e^b=egi,d^c=dg,e^c=ei,f^c=fg,e^d=ei,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44020);