Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4)(6,9)(7,12)(10,11)(14,17)(15,19)(18,20);(5,6)(7,10)(8,11)(9,12)(14,18)(17,20);(13,14,19,20,16,18,15,17);(6,10)(7,12)(9,11)(14,18)(17,20)Polycyclic presentation:
2^9|a^2=g,c^2=fi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72279250, #72279454, #72279630, … |
| maximal normal subgroup of | #72279250, #72279454, #72279630, … |
| maximal quotient of | #72309648, #72309953, #72342483 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132836);