Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
2006 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)(7,11)(10,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,30,33,34)(31,37,38,35)(32,36,40,41)(39,43,44,42);(6,9)(10,12)(14,18)(16,25)(19,21)(22,26)(30,34)(31,38)(32,40)(42,43);(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,25)(26,27)(29,31,33,38)(30,35,34,37)(32,39,40,44)(36,42,41,43);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,15,17,23)(14,19,18,21)(16,26,25,22)(20,28,24,27)(29,30)(31,37)(32,41)(33,34)(35,38)(36,40)(39,42)(43,44);(13,16)(14,20)(15,22)(17,25)(18,24)(19,27)(21,28)(23,26)(29,32)(30,36)(31,39)(33,40)(34,41)(35,42)(37,43)(38,44)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2980873);