Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1957 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)(5,6,8,10)(7,11,12,9)(13,14)(15,21)(16,22)(17,23)(18,20)(19,24)(25,38)(26,37)(27,34)(28,35)(29,39)(30,40)(31,41)(32,36)(33,42)(43,44);(5,7,8,12)(6,9,10,11)(14,20)(15,17)(16,28)(19,26)(21,36)(23,34)(25,43)(27,32)(29,30)(38,39)(40,44)(41,42);(2,4)(13,15)(14,21)(16,29)(17,26)(18,27)(19,32)(20,34)(22,39)(23,37)(24,36)(25,33)(28,43)(30,31)(35,44)(38,42)(40,41);(13,16,18,28)(14,22,20,35)(15,25,27,30)(17,29,32,43)(19,33,26,31)(21,38,34,40)(23,39,36,44)(24,42,37,41);(13,17,18,32)(14,23,20,36)(15,26,27,19)(16,30,28,25)(21,37,34,24)(22,40,35,38)(29,31,43,33)(39,41,44,42)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6481700);