Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478479 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17150 |
| 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 | |
808 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)(3,15,20,27)(4,21)(5,25)(6,12)(7,13)(8,14)(9,18,28,19)(10,23)(11,24)(16,31,17,26)(22,29,32,30);(1,3,8,20)(2,9)(4,22,24,32)(5,16,23,17)(6,18,7,19)(10,29)(11,26)(12,15)(13,27)(14,28)(21,31)(25,30);(1,4)(2,10)(3,16)(5,6)(7,23)(8,24)(9,26)(11,13)(12,21)(14,25)(15,29)(17,20)(18,32)(19,22)(27,30)(28,31);(1,5,7,24,8,23,6,4)(2,11,13,10,14,21,12,25)(3,17,19,22,20,16,18,32)(9,26,27,29,28,31,15,30)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,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385676);