Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23882 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(7,14)(8,9)(11,15)(12,18)(16,17)(19,20);(1,3,4,2)(5,6)(7,15)(8,12)(9,18)(10,13)(11,17)(14,19)(16,20);(5,7,9,17,10,16,8,14)(6,11,12,15,13,19,18,20);(5,6)(7,11)(8,18)(9,12)(10,13)(14,20)(15,17)(16,19)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72221399, C4×#23882, #72252462, … |
| maximal normal subgroup of | C4×#23882, #72252462, #72252536, … |
| maximal quotient of | #72221399, #72234294, #72257935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275285);