Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34911 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26)(27,34)(28,32)(29,37)(30,33)(31,35)(36,39)(38,40);(2,5)(3,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,40)(34,37);(2,5)(6,8)(10,17)(11,20)(16,22)(19,21)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34);(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(26,33)(31,40)(32,37)(34,39)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6565062);