Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14)(15,19)(16,26)(17,21)(18,22)(20,23)(24,28)(25,27);(5,7,8,12)(6,9,10,11)(13,15,18,25)(14,19,22,27)(16,17,23,24)(20,28,26,21);(2,4)(15,23)(16,25)(17,24)(19,20)(21,28)(26,27);(13,16,18,23)(14,20,22,26)(15,24,25,17)(19,21,27,28);(15,16)(17,24)(19,26)(20,27)(21,28)(23,25)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72307120, #72307149, #72307587, … |
| maximal normal subgroup of | #72307120, #72307149, #72307587, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481679);