Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #461277 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16385 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
828 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)(3,15,18,26)(4,19)(5,23)(6,12)(7,13)(8,14)(9,17,27,24)(10,21)(11,22)(16,30,32,25)(20,28,31,29);(1,3,8,18)(2,9)(4,20,22,31)(5,16,21,32)(6,17,7,24)(10,28)(11,25)(12,15)(13,26)(14,27)(19,30)(23,29);(1,4)(2,10)(3,16)(5,6)(7,21)(8,22)(9,25)(11,13)(12,19)(14,23)(15,28)(17,31)(18,32)(20,24)(26,29)(27,30);(1,5,7,22,8,21,6,4)(2,11,13,10,14,19,12,23)(3,16,24,31,18,32,17,20)(9,25,26,28,27,30,15,29)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 368474);