Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456739 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16285 |
| 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 | |
958 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,11,5,18)(3,12,15,8,13,25,19,21)(4,9,16,26,17,24,20,23)(14,29,30,32,28,27,31,22);(1,3,7,15,6,13,5,19)(2,8,10,25,11,21,18,12)(4,14,16,30,17,28,20,31)(9,22,26,29,24,32,23,27);(3,13)(4,16)(5,7)(8,21)(9,23)(10,18)(14,30)(17,20)(22,27)(24,26)(28,31)(29,32);(1,4)(2,9)(3,14)(5,20)(6,17)(7,16)(8,22)(10,26)(11,24)(12,27)(13,28)(15,30)(18,23)(19,31)(21,32)(25,29)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363935);