Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×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 | C22×#493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2946 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)(5,6)(7,8)(9,10,12,11)(14,17)(15,18)(19,20);(1,2,4,3)(5,6)(7,8)(9,11,12,10);(6,8)(13,14,18,20,16,19,15,17);(1,3,4,2)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=fgi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg116 conjugacy classes — view the full character table.
| maximal subgroup of | #72224884, #72224950, #72226806, … |
| maximal normal subgroup of | #72224884, #72224950, #72226806, … |
| maximal quotient of | #72224879, #72309582 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 276261);