Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #114812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #60750 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,7,11,23)(3,13)(4,17)(8,25,27,26)(9,14,12,24)(10,21,28,22)(15,31)(16,30)(18,19)(29,32);(1,2)(3,14)(4,8)(5,12)(6,9)(7,16)(10,18)(11,20)(13,23)(15,22)(17,27)(19,28)(21,32)(24,30)(25,29)(26,31);(2,8)(3,13)(4,18)(7,21)(9,10)(11,27)(12,28)(14,25)(15,32)(16,30)(17,19)(22,23)(24,26)(29,31)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=ghi,d^c=dh,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72213205, #72213613, #72213659, … |
| maximal normal subgroup of | #72213205, #72213613, #72213659, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22008);