Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13673 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator | #127 | C22×C4 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | C2×#3291 | C2 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
887 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 887 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,14,15,10)(8,12,18,11)(9,13)(16,19,17,20);(5,7,9,16)(6,10,13,20)(8,17,18,15)(11,19,12,14);(2,4)(5,6)(7,10)(8,11)(9,13)(12,18)(14,15)(16,20)(17,19)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,d^b=dhi,e^b=egh,f^b=fgh,g^b=gi,d^c=dg,g^c=gi,h^c=hi,e^d=ehi,f^d=fhi,g^e=gi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72279353, #72282060, #72282068, … |
| maximal normal subgroup of | #72279353, #72282060, #72282068, … |
| maximal quotient of | #72283021, #72283238, #72283288, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59382);