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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1838 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)(7,10)(8,9)(11,12)(13,14,17,20)(15,18,21,25)(16,23,24,19)(22,28,27,26)(30,31)(32,39)(36,38)(40,44);(5,7)(6,10)(8,11)(9,12)(15,21)(18,25)(22,27)(26,28)(29,30,34,31)(32,37,39,35)(33,38,41,36)(40,42,44,43);(5,7,9,12)(6,11,8,10)(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2,3,4)(5,7)(6,11)(8,10)(9,12)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,24)(27,28)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(5,8,9,6)(7,10,12,11)(13,16,17,24)(14,19,20,23)(15,22,21,27)(18,26,25,28)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=g,d^2=fhi,e^2=fh,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, 4991628);