Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1866 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)(5,6,9,10)(7,8,11,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,30,34,35)(31,36,40,43)(32,33,41,37)(38,42,44,39);(5,7,9,11)(6,8,10,12)(16,24)(19,26)(22,28)(25,27)(30,35)(33,37)(36,43)(39,42);(6,10)(7,11)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,31,34,40)(30,36,35,43)(32,38,41,44)(33,42,37,39);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,32)(30,37)(31,38)(33,35)(34,41)(36,39)(40,44)(42,43);(5,8)(6,7)(9,12)(10,11)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)(29,33,34,37)(30,32,35,41)(31,39,40,42)(36,44,43,38)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5382668);