Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(9,10,11,12)(14,18)(15,20)(16,19);(1,2)(3,4)(6,8);(5,6,7,8)(13,14,19,20,17,15,16,18);(1,3,4,2)(13,15,16,20,17,14,19,18)Polycyclic presentation:
2^9|a^2=h,c^2=fgi,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239754, #72239763, #72239775 |
| maximal normal subgroup of | #72239754, #72239763, #72239775 |
| maximal quotient of | #72257540 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 294006);