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) = C27 |
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 | |
1921 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)(5,6,8,9)(7,11,12,10)(16,23)(20,25)(21,27)(26,28)(30,34)(32,39)(35,40)(38,43);(6,9)(10,11)(14,18)(19,24)(20,25)(26,28)(30,34)(31,37)(36,41)(38,43);(5,7,8,12)(6,10,9,11)(13,14)(15,19)(16,20)(17,18)(21,26)(22,24)(23,25)(27,28)(29,30)(31,35)(32,36)(33,34)(37,40)(38,42)(39,41)(43,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,31)(30,35)(32,38)(33,37)(34,40)(36,42)(39,43)(41,44);(5,7)(6,10)(8,12)(9,11)(13,16)(14,20)(15,21)(17,23)(18,25)(19,26)(22,27)(24,28)(29,32,33,39)(30,36,34,41)(31,38,37,43)(35,42,40,44)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=fgh,e^2=fg,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, 4516152);