Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1781 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)(8,9)(10,12)(13,14,18,15)(16,21,23,19)(17,22,25,20)(24,26,28,27)(31,37)(32,39)(34,40)(35,42);(5,6)(7,11)(8,9)(10,12)(14,15)(16,23)(20,22)(24,28)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(6,9)(10,11)(13,15,18,14)(16,19,23,21)(17,20,25,22)(24,27,28,26)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27)(29,31)(30,34)(32,38)(33,37)(35,41)(36,40)(39,43)(42,44);(5,7)(6,10)(8,12)(9,11)(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,32,33,39)(30,35,36,42)(31,38,37,43)(34,41,40,44)Polycyclic presentation:
2^9|a^2=f,b^2=fgh,c^2=f,d^2=fghi,e^2=gh,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, 4992130);