Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62243 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62793 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8,6,3,4,7)(9,10)(11,20)(12,17)(13,18)(14,27)(15,19)(16,22)(21,35)(23,30)(24,32)(25,33)(26,34)(28,31)(29,37)(36,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,21)(13,22)(14,28)(15,23)(17,29)(18,30)(19,31)(20,27)(24,36)(25,37)(26,38)(32,35)(33,39)(34,40);(1,4,6,5)(2,7,3,8)(9,12,15,26)(10,17,19,34)(11,21,23,38)(13,25,14,24)(16,29,31,40)(18,33,27,32)(20,35,30,39)(22,37,28,36);(1,4,6,5)(2,7,3,8)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,24,26,25)(16,30,31,20)(17,32,34,33)(21,36,38,37)(29,35,40,39);(1,4,6,5)(2,8,3,7)(9,13,15,14)(10,18,19,27)(11,22,23,28)(12,25,26,24)(16,30,31,20)(17,33,34,32)(21,37,38,36)(29,39,40,35)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=dhi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250436);