Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #352174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#1721 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,13)(7,14,12,9)(8,11,16,18)(15,20,19,17)(21,22)(23,29)(24,32)(25,27)(26,31)(28,33)(30,35)(34,36);(5,7,9,6,10,12,14,13)(8,15,17,11,16,19,20,18)(23,30)(26,34)(29,35)(31,36);(5,8)(6,11)(7,15)(9,17)(10,16)(12,19)(13,18)(14,20)(21,23,24,31,25,30,33,36)(22,26,28,29,27,34,32,35);(1,2,3,4)(8,16)(11,18)(15,19)(17,20)(22,27)(26,34)(28,32)(29,35)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ci,d^c=dh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260188);