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 | |
1692 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)(5,6,10,8)(7,9,12,11)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,34,35)(31,32,38,36)(33,37,42,43)(39,40,41,44);(5,7,10,12)(6,11,8,9)(15,22)(16,25)(18,21)(19,24)(30,35)(31,38)(37,43)(39,41);(6,8)(9,11)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,31,34,38)(30,36,35,32)(33,41,42,39)(37,40,43,44);(1,2)(3,4)(5,8,10,6)(7,9,12,11)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,32,34,36)(30,31,35,38)(33,40,42,44)(37,39,43,41);(5,9)(6,7)(8,12)(10,11)(13,16)(14,19)(15,23)(17,25)(18,26)(20,24)(21,27)(22,28)(29,33)(30,37)(31,39)(32,40)(34,42)(35,43)(36,44)(38,41)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fgh,d^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, 5382697);