Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #435786 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15538 |
| 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 | |
1019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,9,5,11)(3,13,17,8,16,22,14,24)(4,10,19,27,20,25,21,28)(15,26,18,31,29,32,30,23);(1,3)(2,8)(4,18)(5,17)(6,14)(7,16)(9,24)(10,26)(11,22)(12,13)(15,19)(20,30)(21,29)(23,27)(25,32)(28,31);(2,9)(3,14)(5,6)(8,22)(10,25)(13,24)(15,18)(16,17)(19,21)(23,32)(26,31)(29,30);(1,4)(2,10,9,25)(3,15,16,29)(5,19)(6,21)(7,20)(8,23)(11,27,12,28)(13,26)(14,18,17,30)(22,32)(24,31)Polycyclic presentation:
2^9|a^2=g,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342982);