Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6,9,14,10,13,8,18)(7,15,16,12,17,11,19,20);(1,2,3,4)(5,7)(6,11)(8,19)(9,16)(10,17)(12,14)(13,15)(18,20);(5,7)(6,12)(8,19)(9,16)(10,17)(11,14)(13,20)(15,18)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,e^b=ehi,f^b=fh,d^c=dh,e^c=ei,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213329, #72235402, #72235423, … |
| maximal normal subgroup of | #72213329, #72235402, #72235423, … |
| maximal quotient of | #72210462, #72210473, #72231794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53850);