Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #374120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14134 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60369 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2455 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,12,6,10)(3,13,16,23,17,26,15,8)(4,18,21,25,20,24,19,9)(14,22,28,27,29,32,30,31);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,22)(10,26)(11,13)(12,23)(18,27)(19,30)(20,29)(21,28)(24,31)(25,32);(1,2,7,12)(3,8)(4,9,20,25)(5,10,6,11)(13,15)(14,22)(16,26)(17,23)(18,21,24,19)(27,30)(28,31)(29,32);(1,4)(2,9)(3,14)(5,21)(6,19)(7,20)(8,22)(10,24)(11,18)(12,25)(13,27)(15,30)(16,28)(17,29)(23,32)(26,31)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72240440, #72240447, #72240455, … |
| maximal normal subgroup of | #72240440, #72240447, #72240455, … |
| maximal quotient of | #72251229, #72255820, C2×#374123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281319);