Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #391 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1985 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,7,9,15,26,18,13,16,12,19,10,28,27,29)(8,20,23,21,24,34,36,33,25,32,30,35,22,14,31,17);(2,4)(5,7,12,15,9,18,27,16,13,19,11,28,10,29,26,6)(8,21,23,14,24,33,36,20,25,35,30,34,22,17,31,32);(5,8,10,22,13,25,9,24)(6,14,15,20,16,34,28,32)(7,17,18,21,19,33,29,35)(11,23,27,31,12,30,26,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=i,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,g^a=gi,c^b=ch,e^b=ei,f^b=fi,d^c=di,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72237207, #72238782, #72238790, … |
| maximal normal subgroup of | #72237207, #72238782, #72238790, … |
| maximal quotient of | #72237209 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54177);