Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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×#3038 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,14,17,19,18,15,16,20);(5,7,8,6,10,12,9,11)(13,15)(14,18)(16,20)(17,19);(2,4)(13,14)(15,18)(16,19)(17,20);(5,8,10,9)(6,7,11,12)(13,16,18,17)(14,19,15,20)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,d^2=hi,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dh,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257446, #72280072, #72280136, … |
| maximal normal subgroup of | #72257446, #72280072, #72280136, … |
| maximal quotient of | #72257660 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105013);