Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1820 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)(7,11)(10,12)(13,14,17,18)(15,19,21,25)(16,20,24,23)(22,26,27,28)(29,30)(31,34)(32,35)(33,36)(37,41)(38,42)(39,40)(43,44);(6,9)(10,12)(14,18)(15,21)(20,23)(22,27)(29,30,33,36)(31,34,38,42)(32,35,40,39)(37,41,44,43);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27)(29,30)(31,34)(32,39)(33,36)(35,40)(37,43)(38,42)(41,44);(1,2)(3,4)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27)(29,31,33,38)(30,34,36,42)(32,37,40,44)(35,41,39,43);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28)(29,32,33,40)(30,35,36,39)(31,37,38,44)(34,41,42,43)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4518726);