Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509105 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #30396 |
| 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 | |
666 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)(3,14)(4,12)(5,10)(6,23)(7,13)(8,16)(9,15)(11,22)(17,26)(18,27)(19,29)(20,32)(21,28)(24,31)(25,30);(1,3)(2,8)(4,19)(5,17)(6,24)(7,18)(9,28)(10,14)(11,27)(12,31)(13,26)(15,20)(16,22)(21,25)(23,30)(29,32);(1,3)(2,9)(4,15)(5,16)(6,25)(7,18)(8,12)(10,29)(11,30)(13,31)(14,23)(17,22)(19,20)(21,24)(26,28)(27,32);(1,4)(2,10)(3,15)(5,6)(7,21)(9,29)(11,13)(14,27)(16,25)(17,19)(18,24)(20,22)(23,32)(30,31)Polycyclic presentation:
2^9|e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fh,h^c=hi,e^d=ei,f^d=fh,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416301);