Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24397 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,20,26,36)(16,27,29,37)(17,22,18,31)(21,32,38,28)(24,39,41,44)(25,34,30,35)(33,40,43,42);(1,2)(3,4)(6,9)(7,11)(13,15)(14,20)(16,24)(17,30)(18,25)(19,26)(21,33)(22,35)(23,36)(27,39)(28,42)(29,41)(31,34)(32,40)(37,44)(38,43);(2,4)(13,16)(14,21)(15,24)(17,28)(18,32)(19,29)(20,33)(22,37)(23,38)(25,40)(26,41)(27,31)(30,42)(34,39)(35,44)(36,43);(6,9)(10,12)(13,17,19,18)(14,22,23,31)(15,25,26,30)(16,28,29,32)(20,34,36,35)(21,37,38,27)(24,40,41,42)(33,39,43,44);(2,4)(5,7,8,11)(6,10,9,12)(16,29)(21,38)(24,41)(27,37)(28,32)(33,43)(39,44)(40,42)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6580442);