Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #110112 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1437 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
883 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 883 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 3 generators on 32 points:
(1,2,5,11,7,12,4,19)(3,14,18,25,15,30,17,29)(6,13,21,10,22,28,20,9)(8,23,26,31,27,32,24,16);(2,8,12,27)(3,15)(4,20)(5,21)(9,24,10,26)(11,25,19,29)(13,14,28,30)(16,31)(17,23)(18,32);(2,9,28,11)(3,16)(4,20)(5,21)(8,24,30,25)(10,13,19,12)(14,29,27,26)(15,31)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72237938, #72237940, #72237976, … |
| maximal normal subgroup of | #72237938, #72237940, #72237976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17314);