Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163726 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,16,13,15,19,22)(11,14,17,21,18,20,23,24)(25,26)(27,34)(28,32)(29,31)(30,35)(33,37)(36,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11,13,18)(10,14,15,20)(12,17,19,23)(16,21,22,24)(25,27)(26,30)(28,35)(29,36)(31,39)(32,34)(33,38)(37,40);(3,8)(5,7)(10,15)(12,19)(14,20)(17,23)(26,31)(27,35)(29,37)(30,38)(32,33)(34,39);(2,6)(5,7)(26,32)(27,35)(31,33)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=eh,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 122433);