Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D16 |
| commutator | C16 | C2×C42 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 122 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)(3,4)(5,6,8,7)(10,16)(11,12)(13,21)(14,27)(17,29)(18,30)(19,36)(20,35)(22,38)(23,26)(24,25)(28,37)(31,34)(32,39)(33,40);(2,4)(9,10,12,18,13,19,24,35,27,34,38,40,37,39,23,29,15,17,26,32,28,33,22,31,14,20,25,36,21,30,11,16);(5,7,8,6)(10,17)(16,29)(18,32)(19,33)(20,34)(30,39)(31,35)(36,40)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=gh,f^2=g,g^2=hi,h^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gh,h^a=hi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286481, #72286684, #72286704, … |
| maximal normal subgroup of | #72286481, #72286684, #72286704, … |
| maximal quotient of | #72286491, #72321140 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59155);