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 | |
1786 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)(7,11)(10,12)(14,18)(15,21)(20,25)(23,27)(29,30,34,37)(31,32,39,35)(33,36,42,44)(38,40,41,43);(6,9)(10,12)(16,24)(20,25)(23,27)(26,28)(29,31,34,39)(30,35,37,32)(33,41,42,38)(36,40,44,43);(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27)(29,31)(30,35)(32,37)(33,41)(34,39)(36,40)(38,42)(43,44);(1,2,3,4)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,32,34,35)(30,31,37,39)(33,40,42,43)(36,38,44,41);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=fgi,e^2=fh,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, 5188726);