Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #128057 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#342 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
930 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 930 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,22,24,23);(1,2)(3,5,7,8)(4,6)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=i,b^a=bd,c^a=cg,d^a=dg,f^a=fhi,d^b=dh,e^b=egi,h^b=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72231140, #72231165, #72231188, … |
| maximal normal subgroup of | #72231140, #72231165, #72231188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35281);