Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,3,4)(9,10,15,20)(11,16,23,32)(12,24,27,35)(13,19,14,29)(17,25,34,26)(18,28,31,22)(21,33,37,39)(30,38,40,36);(1,2)(3,4)(5,6)(7,8)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,30)(18,29)(19,31)(20,32)(24,33)(25,38)(26,36)(27,37)(34,40)(35,39);(6,8)(9,12,15,27)(10,17,20,34)(11,21,23,37)(13,26,14,25)(16,30,32,40)(18,33,31,39)(19,35,29,24)(22,38,28,36);(2,4)(9,11)(10,18)(12,21)(13,22)(14,28)(15,23)(16,29)(17,33)(19,32)(20,31)(24,30)(25,36)(26,38)(27,37)(34,39)(35,40);(9,13,15,14)(10,19,20,29)(11,22,23,28)(12,25,27,26)(16,18,32,31)(17,24,34,35)(21,36,37,38)(30,39,40,33)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408178);