Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1778 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)(5,6,8,10)(7,9,11,12)(13,14)(15,18)(16,21)(17,20)(19,24)(22,26)(23,28)(25,27)(29,30)(31,37)(32,40)(33,36)(34,39)(35,42)(38,43)(41,44);(7,11)(9,12)(14,15)(19,21)(20,22)(27,28)(29,30,33,36)(31,37,39,34)(32,35,42,40)(38,44,41,43);(5,7)(6,9)(8,11)(10,12)(13,15,18,14)(16,19,24,21)(17,22,26,20)(23,27,25,28)(29,30)(31,37)(32,35)(33,36)(34,39)(38,44)(40,42)(41,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,31,33,39)(30,34,36,37)(32,41,42,38)(35,44,40,43);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,32,33,42)(30,35,36,40)(31,38,39,41)(34,43,37,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2668036);