Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C2×#3035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2740 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,4)(5,6,8,7)(9,10)(11,21)(12,18)(13,19)(14,17)(15,20)(16,25)(22,30)(23,37)(24,35)(26,40)(27,34)(28,32)(29,36)(31,39)(33,38);(6,7)(9,11)(10,16)(12,26)(13,24)(14,25)(15,22)(17,21)(18,33)(19,30)(20,35)(23,27)(28,38)(29,39)(31,34)(32,40)(36,37);(1,2,3,4)(5,6,8,7)(10,17)(11,22)(13,15)(16,30)(18,32)(21,35)(23,26)(24,25)(27,29)(31,40)(33,37)(38,39);(5,7,8,6)(9,12,14,28)(10,18,17,32)(11,23,25,39)(13,27,15,29)(16,31,21,37)(19,34,20,36)(22,26,24,38)(30,40,35,33);(9,13,14,15)(10,19,17,20)(11,24,25,22)(12,27,28,29)(16,30,21,35)(18,34,32,36)(23,38,39,26)(31,40,37,33)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351390);