Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493638 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| prime radical | #63066 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,3,9)(4,10,13,24,16,25,19,26)(5,8,14,20,18,22,17,23)(15,21,28,32,29,31,27,30);(2,8)(3,6)(9,23)(10,21)(11,20)(12,22)(13,19)(14,17)(24,32)(25,31)(26,30)(27,28);(1,3,7,6)(2,9,12,11)(4,13,16,19)(5,17,18,14)(8,20,22,23)(10,24,25,26)(15,28,29,27)(21,30,31,32);(1,4,7,16)(2,10,12,25)(3,13,6,19)(5,15,18,29)(8,21,22,31)(9,24,11,26)(14,27,17,28)(20,30,23,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,c^b=ci,f^b=fhi,g^b=gh,e^c=eh,g^d=gi,f^e=fi,g^e=gi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400848);