Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(9,10)(11,15)(12,13)(14,16)(18,22)(20,23)(25,26)(27,30)(28,33)(29,32)(31,35)(34,37)(36,39)(38,40);(1,2,5,3)(4,7,8,6)(9,11,14,12)(10,13,16,15)(17,18,21,22)(19,23,24,20)(26,27)(28,35)(29,37)(38,39);(9,11,14,12)(10,15,16,13)(17,19)(18,20)(21,24)(22,23)(25,27,30,26)(28,33,35,31)(29,34,37,32)(36,39,40,38);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,13,16,15)(17,20)(18,19)(21,23)(22,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,4,5,8)(2,6,3,7)(9,13)(10,11)(12,16)(14,15)(17,18)(19,20)(21,22)(23,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|b^2=fg,c^2=fgi,d^2=fh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1983015);