Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(5,6,8,7)(9,10)(11,15,16,13)(12,14);(1,2)(3,4)(5,7,8,6)(9,11)(10,13)(12,16)(14,15);(2,4)(9,10)(11,13)(12,14)(15,16);(9,11)(10,13)(12,16)(14,15)Polycyclic presentation:
2^9|a^2=h,b^2=fh,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72261079, #72261088, #72261097, … |
| maximal normal subgroup of | #72261079, #72261088, #72261097, … |
| maximal quotient of | #72312071, #72312615, #72313249 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272319);