Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #110864 |
| 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 | |
875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,19,25,4,9)(3,13,17,29,31,24,16,22)(6,12,20,27,7,11,18,28)(8,21,23,14,30,32,26,15);(2,8,11,24)(3,14)(4,18)(5,20)(9,22,27,23)(10,29,28,26)(12,13,25,30)(15,31)(16,17)(21,32);(2,9,25,10)(3,15)(8,22,30,29)(11,27,12,28)(13,26,24,23)(14,31)(16,32)(17,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,e^2=hi,f^2=hi,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=dh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72232999, #72233128, #72233133, … |
| maximal normal subgroup of | #72232999, #72233128, #72233133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18082);