Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393934 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4)(9,10)(11,12,14,16)(13,15);(5,6)(7,8)(9,11)(10,12)(13,14)(15,16);(2,4)(6,8)(9,12)(10,14)(11,15)(13,16);(9,12)(10,11)(13,16)(14,15)Polycyclic presentation:
2^9|a^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,h^b=hi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225692, #72240106, #72240126, … |
| maximal normal subgroup of | #72225692, #72240106, #72240126, … |
| maximal quotient of | #72225622, #72249561, #72249563, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302614);