Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359528 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3857 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,7,17,20)(3,8,13,22,12,19,27,30)(4,14,16,24,15,9,18,23)(11,21,26,31,25,29,28,32);(2,7)(5,17)(9,14)(13,27)(16,18)(22,30)(26,28)(31,32);(1,3)(2,8)(4,11)(5,13)(6,12)(7,19)(9,21)(10,22)(14,29)(15,25)(16,26)(17,27)(18,28)(20,30)(23,31)(24,32);(1,4)(2,9)(3,11)(5,18)(6,15)(7,14)(8,21)(10,24)(12,25)(13,28)(16,17)(19,29)(20,23)(22,32)(26,27)(30,31)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72238012, #72238015, #72241303, … |
| maximal normal subgroup of | #72238012, #72238015, #72241303, … |
| maximal quotient of | C2×#359529 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266725);