Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,21)(16,29)(17,32)(18,24)(19,36)(20,25)(22,31)(23,33)(26,44)(27,39)(28,41)(30,38)(34,42)(35,40)(37,43);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,20,28)(14,21,25,41)(16,17,26,34)(18,35,19,27)(22,23,38,43)(24,40,36,39)(29,32,44,42)(30,37,31,33);(2,4)(13,16,20,26)(14,22,25,38)(15,17,28,34)(18,31,19,30)(21,23,41,43)(24,44,36,29)(27,37,35,33)(32,40,42,39);(6,10)(7,12)(13,17)(14,23)(15,26)(16,28)(18,33)(19,37)(20,34)(21,38)(22,41)(24,42)(25,43)(27,31)(29,40)(30,35)(32,36)(39,44);(13,18,20,19)(14,24,25,36)(15,27,28,35)(16,30,26,31)(17,33,34,37)(21,39,41,40)(22,29,38,44)(23,42,43,32)Polycyclic presentation:
2^9|a^2=f,b^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, 7408317);