Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99947 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(5,6,9,7,10,12,8,11)(13,14,17,22,18,19,16,21)(15,23,24,27,25,28,26,20);(1,2)(3,4)(6,7)(8,9)(11,12)(14,19)(16,17)(23,28)(24,26);(2,4)(5,7,8,6,10,11,9,12)(13,15)(14,20,19,27)(16,24)(17,26)(18,25)(21,23,22,28)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257946, #72257950, #72257961, … |
| maximal normal subgroup of | #72257946, #72257950, #72257961, … |
| maximal quotient of | #72308716, #72308751 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 26424);