Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501260 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24424 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#444 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1038 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)(3,14)(4,20)(5,16,25,32)(7,13,8,28)(9,30)(10,31)(11,26,23,27)(15,22)(17,24)(18,29)(19,21);(1,3)(2,9)(4,16)(5,22)(6,20)(7,19)(8,18)(10,25)(11,21)(12,24)(13,15)(14,32)(17,26)(23,29)(27,30)(28,31);(1,2)(3,15)(4,10)(5,23)(6,12)(7,28)(8,13)(9,19)(11,25)(14,22)(16,26)(17,29)(18,24)(20,31)(21,30)(27,32);(1,4)(2,10)(3,16)(5,24)(6,14)(7,21)(8,29)(9,25)(11,19)(12,22)(13,17)(15,26)(18,23)(20,32)(27,31)(28,30)Polycyclic presentation:
2^9|a^2=f,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ehi,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ehi,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408456);