Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2240 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)(7,11)(10,12)(13,14,18,22)(15,23,27,34)(16,20,30,37)(17,32,24,21)(19,26,35,28)(25,39,29,43)(31,44,40,38)(33,42,36,41);(6,9)(7,11)(13,15,24,26,18,27,17,28)(14,19,32,34,22,35,21,23)(16,29,40,41,30,25,31,42)(20,36,44,39,37,33,38,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,25)(17,31)(18,30)(19,33)(21,38)(22,37)(23,39)(24,40)(26,42)(27,29)(28,41)(32,44)(34,43)(35,36);(6,9)(10,12)(15,26)(16,30)(17,24)(19,34)(20,37)(21,32)(23,35)(25,41)(27,28)(29,42)(33,39)(36,43);(5,7,8,11)(6,10,9,12)(15,27)(16,30)(19,35)(20,37)(23,34)(26,28)(31,40)(38,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7050988);