Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1772 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14,19,25)(15,26,30,21)(16,31,29,22)(17,35,20,23)(18,37,27,24)(28,39,38,40)(32,41,36,44)(33,42,34,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,30)(14,21,25,26)(16,32,29,36)(17,28,20,38)(18,34,27,33)(22,41,31,44)(23,39,35,40)(24,43,37,42);(2,4)(6,9)(10,11)(13,16,19,29)(14,22,25,31)(15,27,30,18)(17,34,20,33)(21,24,26,37)(23,43,35,42)(28,36,38,32)(39,41,40,44);(5,7,8,12)(6,10,9,11)(13,17,19,20)(14,23,25,35)(15,28,30,38)(16,33,29,34)(18,36,27,32)(21,39,26,40)(22,42,31,43)(24,44,37,41);(13,18,19,27)(14,24,25,37)(15,29,30,16)(17,36,20,32)(21,22,26,31)(23,44,35,41)(28,34,38,33)(39,42,40,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578029);