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 | |
1828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,10,11)(7,12,9,8)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,30,33,36)(31,34,37,41)(32,39,40,35)(38,44,43,42);(5,7,10,9)(6,8,11,12)(14,18)(19,21)(20,24)(26,27)(31,37)(34,41)(38,43)(42,44);(6,11)(7,9)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28)(29,31)(30,34)(32,38)(33,37)(35,42)(36,41)(39,44)(40,43);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,30)(31,34)(32,35)(33,36)(37,41)(38,42)(39,40)(43,44);(5,9,10,7)(6,8,11,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)(29,32,33,40)(30,35,36,39)(31,38,37,43)(34,42,41,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,d^2=fg,e^2=fgh,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, 4511317);