Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196092 |
| 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 | |
2029 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6,9,12,10,14,8,13)(7,15,17,19,18,20,16,11);(5,7)(6,11,14,19)(8,16)(9,17)(10,18)(12,15,13,20);(1,2,3,4);(6,12)(8,9)(11,15)(13,14)(16,17)(19,20)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^b=gi,e^d=ei,g^d=gi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#1904, #72252457, #72252464, … |
| maximal normal subgroup of | D4×#1904, #72252457, #72252464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300258);