Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359650 |
| 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 | #62842 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3903 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)(3,12)(4,16,18,8)(5,9)(6,11)(7,14)(10,20)(13,27,29,32)(15,26)(17,25,31,24)(19,22)(21,30,23,28);(1,3)(2,7)(4,13)(5,19)(6,15)(8,23)(9,12)(10,26)(11,22)(14,20)(16,21)(17,30)(18,29)(24,32)(25,27)(28,31);(1,4)(2,8)(3,13)(5,17)(6,18)(7,21)(9,24)(10,25)(11,16)(12,27)(14,28)(15,29)(19,30)(20,31)(22,23)(26,32);(2,9)(5,20)(7,12)(8,24)(10,11)(14,19)(16,25)(17,31)(21,27)(22,26)(23,32)(28,30)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^c=hi,e^d=ei77 conjugacy classes — view the full character table.
| maximal subgroup of | #72233772, #72233775, #72233776, … |
| maximal normal subgroup of | #72233772, #72233775, #72233776, … |
| maximal quotient of | #72238626, #72249684, #72251290, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266846);