Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,9,13,10,8,12,17)(7,14,16,20,15,11,18,19);(1,2)(3,4)(5,7)(6,11)(8,14)(9,18)(10,15)(12,16)(13,20)(17,19);(2,4)(5,8,9,17,10,6,12,13)(7,14,18,19,15,11,16,20);(5,7)(6,11)(8,14)(9,18)(10,15)(12,16)(13,20)(17,19)Polycyclic presentation:
2^9|a^2=h,c^2=fh,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72238734, #72252449, #72252454, … |
| maximal normal subgroup of | #72238734, #72252449, #72252454, … |
| maximal quotient of | #72238754, #72343062 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281238);