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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1681 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)(5,6,9,8)(7,10,11,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(30,31)(32,39)(36,38)(40,41);(7,11)(10,12)(14,18)(16,25)(19,21)(22,26)(29,30,34,31)(32,37,39,35)(33,38,42,36)(40,43,41,44);(5,7,9,11)(6,10,8,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,25)(26,27)(29,31,34,30)(32,35,39,37)(33,38,42,36)(40,43,41,44);(1,2)(3,4)(13,15,17,23)(14,19,18,21)(16,26,25,22)(20,28,24,27)(29,32,34,39)(30,35,31,37)(33,41,42,40)(36,44,38,43);(5,8,9,6)(7,12,11,10)(13,16)(14,20)(15,22)(17,25)(18,24)(19,27)(21,28)(23,26)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fh,d^2=f,e^2=fgh,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, 2988390);