Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×C42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C22×#95 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,8,5,6,3,7)(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22);(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24);(2,6)(7,8);(1,3,5,4)(2,7,6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=h,f^2=i,b^a=be,c^a=ci,g^a=gh,f^b=fh128 conjugacy classes — view the full character table.
| maximal subgroup of | #72210843, #72210853, #72210860, … |
| maximal normal subgroup of | #72210843, #72210853, #72210860, … |
| maximal quotient of | #72211105, #72237202, #72239204, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88059);