Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #396 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1439 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10)(11,19)(12,23,26,16)(13,27)(14,18)(15,21)(17,29)(20,36,24,34)(22,35)(25,32,28,33)(30,38,40,37)(31,39);(9,11,13,21,14,22,29,39)(10,15,17,19,18,31,27,35)(12,24,25,38,26,20,28,37)(16,30,32,36,23,40,33,34);(9,12,14,26)(10,16,18,23)(11,20,22,24)(13,28,29,25)(15,30,31,40)(17,33,27,32)(19,34,35,36)(21,38,39,37)Polycyclic presentation:
2^9|a^2=g,b^2=d,c^2=h,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,f^b=fh,d^c=dh,g^c=gh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72266601 |
| maximal normal subgroup of | #72266601 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 24190);