Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150243 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#357 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C4×C16 | C23 |
| prime radical | #33713 | C2 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(10,15)(11,19)(12,21)(13,25)(16,28)(17,31)(18,32)(20,24)(22,38)(23,35)(26,37)(27,29)(30,39)(33,34)(36,40);(2,5)(3,7)(9,10)(11,20)(12,24)(13,27)(14,16)(15,19)(17,21)(18,25)(22,39)(23,36)(26,40)(28,35)(29,38)(30,37)(31,33)(32,34);(10,16)(15,28)(17,30)(18,27)(20,36)(24,40)(29,32)(31,39)Polycyclic presentation:
2^9|a^2=e,d^2=gh,e^2=fgh,f^2=hi,g^2=hi,h^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225449, #72225521, #72225531 |
| maximal normal subgroup of | #72225449, #72225521, #72225531 |
| maximal quotient of | #72225435 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58135);