Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201928 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3109 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(3,10)(4,7)(9,14)(11,12)(13,16)(17,18,21,24,22,26,20,25)(19,27,29,23,30,31,28,32);(1,2,7,14,5,9,4,6)(3,8,10,16,11,15,12,13)(17,19)(18,23,26,32)(20,28)(21,29)(22,30)(24,31,25,27);(1,3)(2,8)(4,12)(5,11)(6,13)(7,10)(9,15)(14,16)(18,24)(20,21)(23,31)(25,26)(27,32)(28,29);(3,11)(8,15)(10,12)(13,16)(17,20,22,21)(18,25,26,24)(19,28,30,29)(23,27,32,31)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72309036, C2×#202083 |
| maximal normal subgroup of | #72309036, C2×#202083 |
| maximal quotient of | C2×#202083, #72309103, #72309104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109279);