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 | |
1789 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)(7,8)(11,12)(13,14,17,18)(15,19,21,25)(16,20,24,23)(22,26,27,28)(29,30,34,37)(31,38,32,35)(33,41,42,36)(39,43,40,44);(6,10)(11,12)(14,18)(15,21)(20,23)(22,27)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44);(5,8,9,7)(6,12,10,11)(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28)(29,30)(31,35)(32,38)(33,36)(34,37)(39,43)(40,44)(41,42)Polycyclic presentation:
2^9|a^2=hi,b^2=fg,c^2=fg,d^2=gh,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, 4521890);