Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478348 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2754 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17142 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
704 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,13)(3,14,16,29,17,8,22,10)(4,9,20,26,19,25,21,27)(15,30,18,28,31,23,32,24);(1,3)(2,8)(4,18)(5,22)(6,16)(7,17)(9,24)(10,13)(11,14)(12,29)(15,20)(19,32)(21,31)(23,26)(25,28)(27,30);(1,4)(2,9,11,25)(3,15)(5,21)(6,20)(7,19)(8,23,14,30)(10,28,29,24)(12,27,13,26)(16,32)(17,31)(18,22);(1,3)(2,10,11,29)(4,15)(5,22)(6,16)(7,17)(8,13,14,12)(9,24,25,28)(18,21)(19,31)(20,32)(23,27,30,26)Polycyclic presentation:
2^9|a^2=g,c^2=e,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,g^b=gi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385544);