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 | |
1937 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)(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,21)(17,20)(19,23)(22,25)(24,28)(26,27)(29,30,34,37)(31,35,32,38)(33,41,42,36)(39,44,40,43);(7,11)(9,12)(14,15)(16,23)(17,25)(27,28)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(5,7)(6,9)(8,11)(10,12)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,27,26,28)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,23)(14,19,15,21)(17,26,25,24)(20,28,22,27)(29,30)(31,38)(32,35)(33,36)(34,37)(39,44)(40,43)(41,42);(13,17,18,25)(14,20,15,22)(16,24,23,26)(19,27,21,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=g,d^2=f,e^2=f,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, 2977507);