Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196143 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×#3055 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1917 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)(3,4)(5,6)(7,13,16,10)(8,17)(9,11)(12,18)(14,19,20,15);(2,4)(5,6)(7,14,16,20)(8,12)(9,11)(10,15,13,19)(17,18);(5,7,8,15,9,16,18,19)(6,10,17,20,11,13,12,14);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | D4×#1904, #72252452, #72252460, … |
| maximal normal subgroup of | D4×#1904, #72252452, #72252460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275984);