Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94483 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7324 |
| commutator | C25 | C42 |
| Frattini, cosocle | #2308 | C22 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ #2308 ◁ G | |
| LECS | G ▷ #2308 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
350 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,11)(4,15,17,8)(5,19)(6,10)(7,12,21,16)(9,20)(13,26)(14,23)(18,24,27,25)(22,28,31,29)(30,32);(1,3,5,13)(2,7,9,22,10,21,19,31)(4,16,18,29)(6,14,20,30)(8,23,24,32,15,11,25,26)(12,27,28,17)Polycyclic presentation:
2^9|a^2=d,b^2=e,e^2=f,b^a=bc,c^a=cf,e^a=eg,f^a=fi,c^b=cg,d^b=df,g^b=gh,h^b=hi,d^c=di,e^c=eh,g^e=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72237826, #72237828, #72237834, … |
| maximal normal subgroup of | #72237826, #72237828, #72237834, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1679);