Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #366002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13984 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1283 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)(3,7,13,21)(4,14,17,8)(5,9,18,24)(6,10)(11,22)(12,26)(15,19)(16,28)(20,27)(23,30)(25,31,29,32);(1,3)(2,7,10,21)(4,15,17,11)(5,12)(6,13)(8,22)(9,20,24,26)(14,19)(16,29,30,25)(18,27)(23,32)(28,31);(1,4)(2,8)(3,11)(5,16)(6,17)(7,19)(9,23)(10,14)(12,25)(13,15)(18,30)(20,31)(21,22)(24,28)(26,32)(27,29);(1,5)(2,9,10,24)(3,12)(4,16)(6,18)(7,20,21,26)(8,23,14,28)(11,25)(13,27)(15,29)(17,30)(19,31,22,32)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=f,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,h^c=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72237749, #72237762, #72237766, … |
| maximal normal subgroup of | #72237749, #72237762, #72237766, … |
| maximal quotient of | #72265869 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273198);