Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2825 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)(5,6,10,9)(7,11,8,12)(13,14,19,24)(15,25,28,38)(16,21,29,35)(17,23,18,22)(20,32,34,27)(26,39,40,44)(30,37,31,36)(33,42,43,41);(5,7,10,8)(6,11,9,12)(13,15,19,28)(14,20,24,34)(16,26,29,40)(17,27,18,32)(21,33,35,43)(22,25,23,38)(30,41,31,42)(36,39,37,44);(2,4)(5,6,10,9)(7,12,8,11)(13,16)(14,21)(15,26)(17,30)(18,31)(19,29)(20,33)(22,36)(23,37)(24,35)(25,39)(27,41)(28,40)(32,42)(34,43)(38,44);(5,8,10,7)(6,12,9,11)(16,29)(21,35)(26,40)(30,31)(33,43)(36,37)(39,44)(41,42);(5,9,10,6)(7,11,8,12)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,30,29,31)(20,25,34,38)(21,36,35,37)(26,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6534463);