Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13595 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
864 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 864 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,14,16,18)(8,12,17,11)(9,13)(10,15,19,20);(1,2,3,4)(5,7,8,16,9,15,17,20)(6,10,12,14,13,18,11,19);(7,15)(10,18)(14,19)(16,20)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=hi,f^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=di,e^b=egh,e^d=ehi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279337, #72279361, #72279451, … |
| maximal normal subgroup of | #72279337, #72279361, #72279451, … |
| maximal quotient of | #72280146, #72286414 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59043);