Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13350 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #10991 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,8,13,7,12,24,30)(3,14,17,19,9,28,27,32)(4,18,21,23,20,16,31,15)(5,22,6,11,25,10,26,29);(1,3)(2,4)(5,23)(6,18)(7,9)(8,27)(10,28)(11,19)(12,20)(13,21)(14,22)(15,25)(16,26)(17,24)(29,32)(30,31);(1,4)(2,9)(3,12)(5,14)(6,19)(7,20)(8,21)(10,23)(11,18)(13,27)(15,22)(16,29)(17,30)(24,31)(25,28)(26,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72267972, #72267973, #72267974, … |
| maximal normal subgroup of | #72267972, #72267973, #72267974, … |
| maximal quotient of | C2×#147766, #72280062 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54962);