Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #187944 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3490 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
3026 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)(3,11,14,7)(4,8)(5,10)(6,9)(12,26,28,31)(13,25,29,19)(15,20)(16,22)(17,23)(18,27,24,32)(21,30);(1,3)(2,7)(4,12)(5,14)(6,13)(8,18)(9,19)(10,11)(15,27)(16,28)(17,29)(20,31)(21,26)(22,24)(23,25)(30,32);(1,4,5,16)(2,8,10,22)(3,12,14,28)(6,15,17,30)(7,18,11,24)(9,20,23,21)(13,27,29,32)(19,31,25,26);(2,9)(7,19)(8,20)(10,23)(11,25)(18,31)(21,22)(24,26)Polycyclic presentation:
2^9|a^2=g,c^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^b=fi,g^b=gh,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72235768, #72235808, #72235820, … |
| maximal normal subgroup of | #72235768, #72235808, #72235820, … |
| maximal quotient of | #72237870, #72237881, #72284804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95141);