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 | |
1661 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)(7,11)(8,9)(10,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,30,35,39)(31,37,32,36)(33,34,44,38)(40,43,42,41);(5,6)(7,11)(8,9)(10,12)(16,24)(19,26)(22,28)(25,27)(29,31,35,32)(30,36,39,37)(33,40,44,42)(34,41,38,43);(6,9)(10,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,32,35,31)(30,37,39,36)(33,40,44,42)(34,41,38,43);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,33)(30,38)(31,40)(32,42)(34,39)(35,44)(36,43)(37,41);(5,7,8,12)(6,10,9,11)(13,16)(14,19)(15,22)(17,24)(18,25)(20,26)(21,27)(23,28)(29,34,35,38)(30,33,39,44)(31,41,32,43)(36,40,37,42)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=f,d^2=fi,e^2=fh,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, 5416945);