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 | C23×C4 | C24 |
| prime radical | #62855 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,8,17,23)(3,7,13,21,12,20,16,25)(4,9,15,24,14,22,18,26)(11,27,29,32,28,19,30,31);(1,3)(2,7,8,20)(4,11)(5,16)(6,12)(9,19,22,27)(10,25,23,21)(13,17)(14,28)(15,30)(18,29)(24,32,26,31);(2,8)(5,17)(9,22)(13,16)(15,18)(21,25)(29,30)(31,32);(1,4)(2,9,8,22)(3,11)(5,18)(6,14)(7,19,20,27)(10,26,23,24)(12,28)(13,30)(15,17)(16,29)(21,32,25,31)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,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^b=hi,h^c=hi,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72319429, #72319453, #72319460, … |
| maximal normal subgroup of | #72319429, #72319453, #72319460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271166);