Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1671 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)(7,11,9,12)(13,14)(15,18)(16,21)(17,20)(19,24)(22,25)(23,28)(26,27)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(5,7,10,9)(6,11,8,12)(14,15)(17,25)(19,21)(23,26)(33,41)(36,40)(38,44)(42,43);(5,8,10,6)(7,11,9,12)(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,28,26,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,7,10,9)(6,12,8,11)(13,16,18,24)(14,19,15,21)(17,26,25,23)(20,28,22,27)(29,32,34,30)(31,35,39,37)(33,40,41,36)(38,42,44,43);(5,9,10,7)(6,11,8,12)(13,17)(14,20)(15,22)(16,23)(18,25)(19,27)(21,28)(24,26)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fgh,d^2=fgh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5545121);