Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #36684 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(10,15)(14,20)(16,22)(21,24)(25,28,29,36)(26,31,32,40)(27,33,34,35)(30,37,38,39);(11,17)(14,20)(18,23)(21,24)(26,32)(27,34)(31,40)(33,35);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540608);