Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470395 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1771 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,31)(24,26)(25,29)(27,30)(28,34)(32,35)(33,36);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,34,27,35,25,33)(22,28,29,31,30,36,24,32);(2,4)(7,13)(10,17)(14,19)(18,20)(21,24)(22,26)(23,32)(25,30)(27,29)(28,34)(31,35)(33,36);(7,13)(10,17)(14,19)(18,20)(21,22)(23,28)(24,25)(26,29)(27,30)(31,34)(32,33)(35,36);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,25,27,26)(22,29,30,24)(23,33,35,34)(28,31,36,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403731);