Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434466 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15502 |
| 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 | |
931 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,13)(3,14,17,30)(4,18)(5,9,16,11)(6,24,7,12)(10,21)(15,27)(19,26)(20,31)(22,29)(23,28)(25,32);(1,3)(2,9)(4,19)(5,7)(6,16)(8,17)(10,26)(11,13)(12,30)(14,24)(15,22)(18,27)(20,23)(21,32)(25,28)(29,31);(1,4)(2,10,13,29)(3,15)(5,20)(6,21)(7,22)(8,23)(9,25,11,27)(12,18,24,28)(14,31,30,26)(16,19)(17,32);(1,5,7,3,8,16,6,17)(2,11,24,14,13,9,12,30)(4,20,22,15,23,19,21,32)(10,27,28,31,29,25,18,26)Polycyclic presentation:
2^9|a^2=eg,c^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341663);