Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125725 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1595 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | #10032 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,20)(12,16)(13,27)(14,19)(15,22)(17,25)(18,26)(21,37)(23,34)(24,38)(28,31)(29,35)(30,33)(32,40)(36,39);(9,11)(10,15,19,32)(12,21)(13,28)(14,24)(16,31,18,36)(17,34,35,37)(20,27,38,33)(22,29)(23,26)(25,40)(30,39);(2,5)(3,8)(6,7)(9,12,14,26)(10,16,19,18)(11,21,24,23)(13,29,30,25)(15,31,32,36)(17,33,35,27)(20,37,38,34)(22,39,40,28)Polycyclic presentation:
2^9|b^2=gi,c^2=i,d^2=fi,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,d^b=dfi,e^b=egi,f^b=fi,d^c=dg,e^c=eh,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72213801 |
| maximal normal subgroup of | #72213801 |
| maximal quotient of | #72213261 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32924);