Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359679 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11)(3,12)(4,17,20,26)(5,7,21,10)(8,16)(9,19,25,18)(13,22)(14,29)(15,24)(23,31)(27,28)(30,32);(2,7)(3,13)(5,21)(9,17)(10,11)(14,28)(15,16)(18,19)(22,24)(25,26)(29,32)(30,31);(1,3)(2,8)(4,14)(5,15)(6,16)(7,22)(9,23)(10,24)(11,12)(13,21)(17,32)(18,28)(19,31)(20,30)(25,27)(26,29);(1,4)(2,9)(3,14)(5,19)(6,20)(7,17)(8,23)(10,26)(11,25)(12,27)(13,28)(15,31)(16,30)(18,21)(22,32)(24,29)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72215974, #72216047, #72216071, … |
| maximal normal subgroup of | #72215974, #72216047, #72216071, … |
| maximal quotient of | #72237508, #72237510, C2×#359679, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266875);