Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1682 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)(8,10)(9,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30,34,38)(31,32,36,35)(33,37,42,44)(39,40,41,43);(5,6)(7,11)(8,10)(9,12)(15,21)(19,22)(23,27)(26,28)(29,31,34,36)(30,35,38,32)(33,39,42,41)(37,43,44,40);(5,6,8,10)(7,9,12,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31)(30,32)(33,41)(34,36)(35,38)(37,43)(39,42)(40,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32)(30,36)(31,38)(33,40)(34,35)(37,41)(39,44)(42,43);(5,7)(6,9)(8,12)(10,11)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33)(30,37)(31,39)(32,40)(34,42)(35,43)(36,41)(38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=g,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5123509);