Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383150 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14357 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2001 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,11,21,10)(3,12,14,23,16,27,15,7)(4,8,20,25,19,26,18,24)(13,29,31,32,17,28,30,22);(1,3)(2,7)(4,17)(5,14)(6,16)(8,22)(9,12)(10,27)(11,23)(13,19)(15,21)(18,31)(20,30)(24,28)(25,29)(26,32);(1,4)(2,8)(3,13)(5,18)(6,19)(7,22)(9,24)(10,25)(11,26)(12,28)(14,30)(15,31)(16,17)(20,21)(23,32)(27,29);(2,9)(3,14)(5,21)(8,24)(10,11)(12,27)(13,30)(15,16)(17,31)(18,20)(25,26)(28,29)Polycyclic presentation:
2^9|a^2=h,e^2=i,h^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,h^c=hi,e^d=ei,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72240435, #72240441, #72240457, … |
| maximal normal subgroup of | #72240435, #72240441, #72240457, … |
| maximal quotient of | #72239777, #72251198, #72251321 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290348);