Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#120 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(2,5)(6,8)(12,22)(14,23)(15,20)(19,21);(11,18)(12,22)(15,20)(17,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^b=fi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257570, #72263766, #72280233, … |
| maximal normal subgroup of | #72257570, #72263766, #72280233, … |
| maximal quotient of | #72264328, #72318087 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361031);