Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1698 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,12)(9,11)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(30,32)(31,37)(36,40)(39,41);(5,7)(6,10)(8,11)(9,12)(16,24)(19,26)(22,28)(25,27)(29,30,34,32)(31,38,37,35)(33,36,42,40)(39,44,41,43);(6,11)(7,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,31,34,37)(30,35,32,38)(33,41,42,39)(36,44,40,43);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,32,34,30)(31,35,37,38)(33,40,42,36)(39,43,41,44);(5,8)(6,7)(9,10)(11,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)(29,33,34,42)(30,36,32,40)(31,39,37,41)(35,43,38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=f,d^2=fhi,e^2=fgh,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, 5418229);