Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,19,22,34)(12,16)(13,26,28,17)(14,18)(15,21)(20,35,23,37)(24,32)(25,40,39,33)(27,30)(29,36)(31,38);(9,11,14,22)(10,15,18,30)(12,23,24,20)(13,27,28,21)(16,31,32,29)(17,34,26,19)(25,36,39,38)(33,35,40,37);(1,3,5,4)(2,6,8,7)(12,24)(16,32)(20,23)(25,39)(29,31)(33,40)(35,37)(36,38);(9,12,14,24)(10,16,18,32)(11,20,22,23)(13,25,28,39)(15,29,30,31)(17,33,26,40)(19,35,34,37)(21,36,27,38)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,f^e=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387315);