Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2213 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
659 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)(6,10)(7,14)(8,13)(11,18)(12,17)(15,20)(16,19);(1,2,4,3)(6,11)(10,18)(12,19)(14,20);(1,3,4,2)(5,6,8,10,9,11,13,18)(7,12,17,14,15,19,16,20);(5,7)(6,12)(8,17)(9,15)(10,14)(11,19)(13,16)(18,20)Polycyclic presentation:
2^9|b^2=fhi,c^2=eh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72226767, C2×#512192 |
| maximal normal subgroup of | #72226767, C2×#512192 |
| maximal quotient of | C2×#512192 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419388);