Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19279 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
936 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(14,19)(15,20)(16,17);(1,3,4,2)(13,14,16,20,18,15,17,19);(6,9)(7,12)(10,11)(13,15,17,20,18,14,16,19);(6,10)(9,11)(13,16,18,17)(14,20,15,19)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,f^c=fhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72279582, #72279595, #72279671 |
| maximal normal subgroup of | #72279582, #72279595, #72279671 |
| maximal quotient of | #72309634, #72309658, #72342622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109997);