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 | |
1808 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)(6,9)(10,12)(13,14,18,19)(15,20,24,27)(16,17,25,21)(22,26,28,23)(29,30)(31,34)(32,40)(33,37)(35,39)(36,42)(38,44)(41,43);(7,11)(10,12)(14,19)(17,21)(20,27)(23,26)(29,31,33,39)(30,34,37,35)(32,41,42,38)(36,44,40,43);(5,6,8,9)(7,10,11,12)(13,15,18,24)(14,20,19,27)(16,22,25,28)(17,26,21,23)(29,31)(30,35)(32,38)(33,39)(34,37)(36,44)(40,43)(41,42);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,22)(17,19)(18,25)(20,23)(24,28)(26,27)(29,30)(31,34)(32,36)(33,37)(35,39)(38,43)(40,42)(41,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,21)(14,16,19,25)(15,23,24,26)(20,28,27,22)(29,32)(30,36)(31,38)(33,42)(34,43)(35,44)(37,40)(39,41)Polycyclic presentation:
2^9|a^2=f,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, 5382648);