Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
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×#385 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
849 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(4,11)(6,12)(7,14)(9,15)(17,18,22,25,23,20,21,26)(19,27,30,24,31,32,29,28);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19,22,30,23,31,21,29)(18,24,25,32,20,28,26,27);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20)(18,23)(19,28)(21,25)(22,26)(24,31)(27,30)(29,32);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,23,22)(18,25,20,26)(19,29,31,30)(24,32,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=i,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99677);