Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,9)(8,10)(11,12)(13,14,17,19,16,15,18,20);(1,2,4,3)(6,9)(7,12)(10,11)(14,15)(19,20);(1,3,4,2)(14,15)(19,20);(6,10)(9,11)(13,15,17,20,16,14,18,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=h,f^2=i,b^a=be,c^a=ci,d^a=dh,e^a=eh,d^b=di,e^b=eh,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72279339, #97×AGL(1,ℤ/8), #72309473, … |
| maximal normal subgroup of | #72279339, #97×AGL(1,ℤ/8), #72309473, … |
| maximal quotient of | #72309489 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91193);