Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437445 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3492 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3326 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)(4,7)(5,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(1,3,5,4)(2,7,8,6)(12,22)(14,23)(15,20)(19,21);(1,4,5,3)(2,7,8,6)(12,22)(14,23)(15,20)(19,21);(11,18)(14,23)(17,24)(19,21)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72257530, #72257550, #72257570, … |
| maximal normal subgroup of | #72257530, #72257550, #72257570, … |
| maximal quotient of | #72264347 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360133);