Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34963 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,22,23,21,25,19,26)(20,28,30,32,29,24,27,31);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,22)(18,23,25,26)(20,27,29,30)(24,31,28,32);(10,14)(11,15)(17,20,22,30,21,29,19,27)(18,24,23,31,25,28,26,32);(2,5)(6,8)(10,14)(13,16)(18,25)(23,26)(24,28)(31,32);(1,3,4,7)(2,6,5,8)(17,19,21,22)(18,23,25,26)(20,27,29,30)(24,31,28,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72284258, #72284418, #72284474, … |
| maximal normal subgroup of | #72284258, #72284418, #72284474, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6319104);