Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388123 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14516 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
949 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)(3,15)(4,19,5,10)(6,13)(7,14)(8,12)(9,17)(11,23,30,21)(16,26,32,28)(18,31)(20,29,22,25)(24,27);(1,3,8,17,7,18,6,24)(2,9,12,31,14,27,13,15)(4,20,21,32,5,22,23,16)(10,28,30,25,19,26,11,29);(1,4,7,5)(2,10)(3,16,18,32)(6,21,8,23)(9,25)(11,12)(13,30)(14,19)(15,26)(17,22,24,20)(27,29)(28,31);(1,5,7,4)(2,11)(3,16,18,32)(6,21,8,23)(9,26)(10,12)(13,19)(14,30)(15,25)(17,20,24,22)(27,28)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=efg,d^2=efg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72251777, #72251788, #72251796, … |
| maximal normal subgroup of | #72251777, #72251788, #72251796, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295319);