Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | C2×#327 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5,6,7)(9,10,14,20,15,22,30,19,16,21,28,35,12,18,13,29)(11,23,26,17,24,33,38,32,27,36,39,34,31,40,25,37);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17,21,34)(12,24)(13,26)(14,25)(15,31)(16,27)(18,32,22,37)(19,33,29,40)(20,36,35,23)(28,38)(30,39);(9,12,16,15)(10,18,21,22)(11,24,27,31)(13,28,30,14)(17,32,34,37)(19,20,29,35)(23,33,36,40)(25,26,38,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fi,d^b=dg,e^b=efgi,g^b=gi,h^b=hi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72213052, #72213801, #72214501, … |
| maximal normal subgroup of | #72213052, #72213801, #72214501, … |
| maximal quotient of | #72211371 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31294);