Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405411 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14912 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,12)(5,19)(8,22)(9,21)(10,11)(14,29)(15,27)(16,17)(24,25)(26,30)(28,31);(1,2)(3,13,15,20)(4,9)(5,10)(6,11)(7,19)(8,12,22,27)(14,28,30,31)(16,24)(17,21)(18,25)(23,26,32,29);(1,3,5,12,6,15,19,27)(2,8,7,20,11,22,10,13)(4,14,16,26,18,30,17,29)(9,23,21,31,25,32,24,28);(1,4)(2,9)(3,14)(5,16)(6,18)(7,21)(8,23)(10,24)(11,25)(12,26)(13,28)(15,30)(17,19)(20,31)(22,32)(27,29)Polycyclic presentation:
2^9|b^2=h,c^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237692, #72240442, #72240455, … |
| maximal normal subgroup of | #72237692, #72240442, #72240455, … |
| maximal quotient of | #72237704 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312607);