Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2380 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13832 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,11,6,22)(3,12,15,8,16,24,14,25)(4,17,19,28,21,9,20,27)(13,29,30,32,31,26,18,23);(1,3,7,16)(2,8)(4,18)(5,14,6,15)(9,26,17,29)(10,12)(11,25)(13,20)(19,31)(21,30)(22,24)(23,27,32,28);(1,2,7,11)(3,8)(4,9,21,17)(5,10,6,22)(12,14)(13,23)(15,24)(16,25)(18,26)(19,27,20,28)(29,30)(31,32);(1,4)(2,9)(3,13)(5,20)(6,19)(7,21)(8,23)(10,28)(11,17)(12,29)(14,30)(15,18)(16,31)(22,27)(24,26)(25,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,h^a=hi,c^b=cef,d^b=de,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284807, #72284828, #72284930, … |
| maximal normal subgroup of | #72284807, #72284828, #72284930, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264567);