Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #110146 |
| 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 | |
843 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 843 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)(3,14,18,27,15,30,17,25)(6,13,21,9,22,28,20,10)(8,23,24,31,26,32,29,16);(2,8,12,26)(3,15)(4,20)(5,21)(9,24,10,29)(11,27,19,25)(13,14,28,30)(16,31)(17,23)(18,32);(2,9,13,11)(3,16)(4,21)(5,20)(8,24,14,27)(10,28,19,12)(15,31)(17,18)(23,32)(25,26,29,30)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=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237938, #72237941, #72237978, … |
| maximal normal subgroup of | #72237938, #72237941, #72237978, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17348);