Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388001 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14507 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
911 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)(3,7,13,22)(4,8,16,25)(5,17)(6,10)(9,19)(11,20)(12,21,18,29)(14,23)(15,24,28,30)(26,32)(27,31);(1,3)(2,7,10,22)(4,14,16,11)(5,18,19,12)(6,13)(8,23)(9,21)(15,26)(17,29)(20,25)(24,32,30,31)(27,28);(1,4)(2,8,10,25)(3,11)(5,15)(6,16)(7,20,22,23)(9,24,17,30)(12,26)(13,14)(18,27)(19,28)(21,31,29,32);(1,5)(2,9,10,17)(3,12)(4,15)(6,19)(7,21,22,29)(8,24,25,30)(11,26)(13,18)(14,27)(16,28)(20,31,23,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=eg,d^2=eg,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237731, #72237732, #72237749, … |
| maximal normal subgroup of | #72237731, #72237732, #72237749, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295197);