Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129617 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #37232 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(5,6)(7,14,16,11)(8,12,17,10)(9,13)(15,19)(18,20);(1,2)(3,4)(6,10)(8,17)(11,19)(12,13)(14,20)(15,18);(2,4)(5,7)(6,11)(8,18)(9,16)(10,19)(12,20)(13,14)(15,17)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,h^a=hi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,d^c=dh,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72232788, #72233009, #72233087, … |
| maximal normal subgroup of | #72232788, #72233009, #72233087, … |
| maximal quotient of | #72232784, #72233006, #72234552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55025);