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 | #62852 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,5,20)(3,13,15,8,16,26,21,25)(4,17,18,10,19,9,22,28)(14,24,31,27,30,32,29,23);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,27)(10,24)(11,26)(12,25)(13,20)(17,23)(18,31)(19,30)(22,29)(28,32);(1,4)(2,9)(3,14)(5,18)(6,22)(7,19)(8,23)(10,11)(12,17)(13,24)(15,29)(16,30)(20,28)(21,31)(25,27)(26,32);(1,4)(2,10)(3,14)(5,22)(6,18)(7,19)(8,24)(9,11)(12,28)(13,23)(15,31)(16,30)(17,20)(21,29)(25,32)(26,27)Polycyclic presentation:
2^9|a^2=h,e^2=i,h^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 | #72233773, #72233776, #72233778, … |
| maximal normal subgroup of | #72233773, #72233776, #72233778, … |
| maximal quotient of | C2×#359652, #72317831, #72318137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266848);