Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433042 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,27,30)(23,32,34,36)(24,33,28,29)(25,31,26,35);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,26,33,27,34,25,29)(22,28,31,32,30,24,35,36);(2,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,31)(25,32)(26,36)(27,28)(30,33)(34,35);(6,11)(10,16)(12,18)(17,20)(22,30)(24,28)(31,35)(32,36);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(21,25,27,26)(22,31,30,35)(23,33,34,29)(24,32,28,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343957);