Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60799 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1337 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,13,17,29,3,15)(4,20,23,27,32,12,18,30)(6,14,16,8,7,10,19,11)(9,26,25,22,28,31,24,21);(2,8,14,13)(3,16)(4,21)(5,19)(9,24,12,27)(10,15,29,11)(20,30,28,25)(22,32);(2,9)(3,16)(4,22)(5,19)(8,24)(10,20)(11,25)(12,14)(13,27)(15,30)(18,23)(21,32)(26,31)(28,29);(2,10)(8,15)(9,20)(11,13)(12,28)(14,29)(24,30)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72213691, #72213874, #72219527, … |
| maximal normal subgroup of | #72213691, #72213874, #72219527, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108252);