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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1861 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)(8,12)(9,10)(13,14,17,18)(15,21,23,19)(16,20,24,26)(22,28,25,27)(29,30)(31,35)(32,34)(33,40)(36,42)(37,39)(38,44)(41,43);(5,7,10,12)(6,8,9,11)(14,18)(16,24)(19,21)(22,25)(30,32)(31,37)(36,40)(38,41);(6,9)(7,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,24)(26,27)(29,31,34,37)(30,35,32,39)(33,41,42,38)(36,44,40,43);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,32,34,30)(31,35,37,39)(33,40,42,36)(38,43,41,44);(5,9,10,6)(7,8,12,11)(13,16)(14,20)(15,22)(17,24)(18,26)(19,27)(21,28)(23,25)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,41)(39,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=fg,d^2=fg,e^2=fgh,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, 4948746);