Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(10,14,15,23)(11,17)(12,20)(16,18,21,24)(19,22);(5,6,7,8)(9,10)(11,18)(12,21)(13,15)(14,22)(16,20)(17,23)(19,24);(1,2,3,4)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,d^b=dhi,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72309811, #72309815, D4×#1386, … |
| maximal normal subgroup of | #72309811, #72309815, D4×#1386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54237);