Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499785 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23848 |
| 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 | |
1255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,9)(3,14)(4,20)(5,24)(6,7)(10,23)(12,32)(13,29)(15,21)(16,28)(17,25)(18,19)(22,30)(26,27);(1,2)(3,15)(4,11)(5,16)(6,12)(7,28)(8,13)(9,24)(10,18)(14,20)(17,22)(19,30)(21,25)(23,31)(26,29)(27,32);(1,3)(2,10)(4,16)(5,25)(6,18)(7,14)(8,19)(9,21)(11,26)(12,30)(13,20)(15,28)(17,27)(22,29)(23,32)(24,31);(1,4)(2,11)(3,16)(5,15)(6,22)(7,21)(8,23)(9,14)(10,26)(12,17)(13,31)(18,29)(19,32)(20,24)(25,28)(27,30)Polycyclic presentation:
2^9|e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=eh,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,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, 406981);