Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19254 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C22×#41 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,8,6,9,4,14)(3,10,12,15,13,16,11,7)(17,18,21,25,22,24,20,29)(19,26,27,23,28,32,30,31);(1,3,4,11,6,13,5,12)(2,7,8,10,9,15,14,16);(17,19)(18,23)(20,30)(21,27)(22,28)(24,31)(25,32)(26,29);(18,24)(23,31)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72321396, #72321416, #72321454, … |
| maximal normal subgroup of | #72321396, #72321416, #72321454, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96455);