Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(14,18)(16,23)(19,24)(22,27)(29,30)(31,37)(32,36)(33,34)(35,38)(39,44)(40,42)(41,43);(1,2,3,4)(15,21)(19,24)(22,27)(26,28)(30,34)(31,38)(36,40)(39,41);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,30)(31,35)(32,40)(33,34)(36,42)(37,38)(39,44)(41,43);(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,31,33,38)(30,35,34,37)(32,41,42,39)(36,44,40,43);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32)(30,36)(31,39)(33,42)(34,40)(35,43)(37,44)(38,41)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=fi,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3454880);