Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2202 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,16,12,14)(18,22)(19,20)(23,24);(11,12)(14,16)(17,18)(19,22)(20,24)(21,23);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16);(9,12,13,11)(10,14,15,16);(17,19,21,20)(18,22,23,24)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257879, #72257886, #72257966, … |
| maximal normal subgroup of | #72257879, #72257886, #72257966, … |
| maximal quotient of | #72307163 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370803);