Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #383745 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60426 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1379 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)(3,9,13,7)(4,15)(5,19)(6,10)(8,17)(11,22)(12,26,30,23)(14,24,20,28)(16,25,31,32)(18,27)(21,29);(1,3)(2,7)(4,16)(5,20)(6,13)(8,25)(9,10)(11,28)(12,18)(14,22)(15,32)(17,31)(19,24)(21,30)(23,27)(26,29);(1,4)(2,8,10,15)(3,12)(5,21)(6,17)(7,23,9,26)(11,29,19,27)(13,30)(14,31)(16,20)(18,22)(24,32,28,25);(1,3)(2,9,10,7)(4,12)(5,14)(6,13)(8,26,15,23)(11,28,19,24)(16,18)(17,30)(20,22)(21,31)(25,27,32,29)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72240435, #72240447, #72240448, … |
| maximal normal subgroup of | #72240435, #72240447, #72240448, … |
| maximal quotient of | #72251310, #72255836 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290941);