Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,10)(7,12)(13,14,18,15)(16,21,23,19)(17,22,25,20)(24,26,28,27)(29,30,33,36)(31,37,39,34)(32,35,40,42)(38,43,44,41);(5,6,9,10)(7,8,12,11)(14,15)(16,23)(20,22)(24,28)(32,40)(35,42)(38,44)(41,43);(7,12)(8,11)(13,15,18,14)(16,19,23,21)(17,20,25,22)(24,27,28,26)(29,31,33,39)(30,34,36,37)(32,38,40,44)(35,41,42,43);(1,2,3,4)(5,7,9,12)(6,11,10,8)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(5,8,9,11)(6,12,10,7)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,32)(30,35)(31,38)(33,40)(34,41)(36,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=gh,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 4989835);