Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(7,11)(13,14,17,21)(15,23,26,19)(16,20,28,36)(18,22,30,38)(24,41,42,33)(25,39,31,34)(27,44,43,35)(29,37,32,40);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,27)(17,26)(18,31)(20,35)(21,23)(22,39)(24,29)(25,30)(28,43)(32,42)(33,37)(34,38)(36,44)(40,41);(1,2,3,4)(13,16)(14,20)(15,24)(17,28)(18,32)(19,33)(21,36)(22,40)(23,41)(25,43)(26,42)(27,31)(29,30)(34,44)(35,39)(37,38);(5,7,8,11)(6,10,9,12)(14,21)(16,28)(19,23)(22,38)(24,42)(27,43)(29,32)(34,39);(6,9)(10,12)(15,25)(18,30)(19,34)(22,38)(23,39)(24,43)(26,31)(27,42)(29,32)(33,44)(35,41)(37,40)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590698);