Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(9,10,15,21)(11,23,27,17)(12,28,25,36)(13,32,24,40)(14,22,16,20)(18,30,37,31)(19,29,38,33)(26,35,34,39);(1,2,4,3)(5,6)(7,8)(9,11)(10,17)(12,29)(13,30)(14,26)(15,27)(16,34)(18,32)(19,28)(20,35)(21,23)(22,39)(24,31)(25,33)(36,38)(37,40);(9,12,15,25)(10,18,21,37)(11,24,27,13)(14,31,16,30)(17,19,23,38)(20,28,22,36)(26,33,34,29)(32,35,40,39);(6,8)(9,13,15,24)(10,19,21,38)(11,25,27,12)(14,33,16,29)(17,18,23,37)(20,32,22,40)(26,31,34,30)(28,35,36,39);(1,3,4,2)(9,14,15,16)(10,20,21,22)(11,26,27,34)(12,30,25,31)(13,33,24,29)(17,35,23,39)(18,36,37,28)(19,32,38,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426389);