Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19279 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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×#388 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(3,10)(4,7)(8,16)(9,14)(11,13)(17,18,20,23,22,25,21,26)(19,27,29,32,28,31,30,24);(1,2,7,14,5,9,4,6)(3,11,15,13,10,8,12,16)(18,23,25,26)(19,28)(24,31,32,27)(29,30);(1,3,4,12,5,10,7,15)(2,8,6,13,9,11,14,16)(17,19,20,29,22,28,21,30)(18,24,26,31,25,32,23,27);(18,25)(23,26)(24,32)(27,31)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,f^c=fhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285957, #72322077, #72322139 |
| maximal normal subgroup of | #72285957, #72322077, #72322139 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 110271);