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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3134 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)(9,10,13,18)(11,20,22,15)(12,17,24,31)(14,16,26,19)(21,33,34,28)(23,29,36,27)(25,30,37,32)(35,39,40,38);(1,2,4,3)(5,6)(7,8)(9,11,13,22)(10,15,18,20)(12,21,24,34)(14,23,26,36)(16,27,19,29)(17,28,31,33)(25,35,37,40)(30,38,32,39);(1,3,4,2)(6,8)(10,16)(14,26)(15,27)(17,30)(18,19)(20,29)(23,36)(25,37)(28,38)(31,32)(33,39)(35,40);(6,8)(9,12)(10,17)(11,21)(13,24)(14,25)(15,28)(16,30)(18,31)(19,32)(20,33)(22,34)(23,35)(26,37)(27,38)(29,39)(36,40);(10,18)(11,22)(12,24)(16,19)(23,36)(25,37)(28,33)(38,39)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346033);