Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196110 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3036 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1903 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,10,14,18)(8,12)(9,11)(13,17,16,20)(15,19);(1,2,4,3)(7,13)(8,15)(10,17)(12,19)(14,16)(18,20);(5,7)(6,10,11,18)(8,16)(9,14)(12,20,19,17)(13,15);(1,3,4,2)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#1904, #72252458, #72252463, … |
| maximal normal subgroup of | D4×#1904, #72252458, #72252463, … |
| maximal quotient of | C2×#374381 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281577);