Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363969 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13956 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #62844 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2689 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)(3,7,13,20)(4,14,16,8)(5,9,17,23)(6,10)(11,18)(12,26)(15,28)(19,27)(21,25)(22,29)(24,31,32,30);(1,3)(2,7,10,20)(4,11)(5,12)(6,13)(8,21,14,18)(9,19,23,26)(15,24)(16,25)(17,27)(22,31,28,30)(29,32);(1,4)(2,8)(3,11)(5,15)(6,16)(7,18)(9,22)(10,14)(12,24)(13,25)(17,29)(19,30)(20,21)(23,28)(26,31)(27,32);(1,5)(2,9,10,23)(3,12)(4,15)(6,17)(7,19,20,26)(8,22,14,28)(11,24)(13,27)(16,29)(18,30,21,31)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,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^b=hi,h^c=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72234953, #72237785, #72237796, … |
| maximal normal subgroup of | #72234953, #72237785, #72237796, … |
| maximal quotient of | #72234957, #72234960, #72238859, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271165);