Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93145 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C25 | C42 |
| Frattini, prime radical, cosocle | C22×#127 | C22 |
| LCS | G ▷ C25 ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C22×#127 ▷ C24 ▷ C22 ▷ C1 | |
720 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 720 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,14)(11,16)(12,15);(2,5)(6,7)(9,11,12,10,13,15,16,14)Polycyclic presentation:
2^9|a^2=d,b^2=e,e^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gi,c^b=cg,d^b=df,f^b=fi,g^b=gi,d^c=dh,e^c=ei,e^d=ei44 conjugacy classes — view the full character table.
| maximal subgroup of | #72210549, #72210563, #72210569, … |
| maximal normal subgroup of | #72210549, #72210563, #72210569, … |
| maximal quotient of | C2×#93146 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342);