Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60734 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,11,12,7)(5,8)(9,13,15,16)(10,14)(18,22,24,27)(19,25,26,28);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,16)(12,15)(17,18,20,27,21,24,29,22)(19,23,28,31,26,32,25,30);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,19)(18,23)(20,28)(21,26)(22,30)(24,32)(25,29)(27,31);(2,8)(6,14)(7,11)(13,16)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102285);