Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232138 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3151 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1766 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 1766 conjugacy classes of subgroups — too many to draw. Only its 266 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 — 4 generators on 20 points:
(2,3)(5,6,10,18,12,17,11,16)(7,14,9,20,8,15,19,13);(1,2)(3,4)(5,7,12,8)(6,13,17,20)(9,11,19,10)(14,16,15,18);(5,8,11,19,12,7,10,9)(6,14,16,20,17,15,18,13);(1,3,4,2)(5,9,10,7,12,19,11,8)(6,15,16,13,17,14,18,20)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^c=hi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279685, #72339817, #72339841, … |
| maximal normal subgroup of | #72279685, #72339817, #72339841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293232);