Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1816 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(10,11)(13,14,17,19,16,15,18,20);(5,6)(7,8)(9,10)(11,12);(2,4)(6,8)(9,11,12,10)(14,15)(19,20);(13,15,17,20,16,14,18,19)Polycyclic presentation:
2^9|a^2=h,c^2=e,d^2=h,h^2=i,b^a=be,c^a=cf,c^b=cg,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72264277, #72264302, #72264338, … |
| maximal normal subgroup of | #72264277, #72264302, #72264338, … |
| maximal quotient of | #72322731 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266269);