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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1860 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)(5,6,8,10)(7,9,11,12)(13,14)(15,21)(16,24)(17,18)(19,22)(20,26)(23,27)(25,28)(29,30)(31,36)(32,39)(33,35)(34,41)(37,42)(38,44)(40,43);(7,11)(9,12)(14,18)(15,22)(20,24)(23,25)(29,31,35,32)(30,36,33,39)(34,38,42,40)(37,43,41,44);(5,7)(6,9)(8,11)(10,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,28)(24,26)(25,27)(29,32,35,31)(30,36,33,39)(34,40,42,38)(37,43,41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,25)(17,22)(18,21)(20,28)(23,26)(24,27)(29,33,35,30)(31,36,32,39)(34,41,42,37)(38,43,40,44);(13,16,17,26)(14,20,18,24)(15,23,22,25)(19,27,21,28)(29,34)(30,37)(31,38)(32,40)(33,41)(35,42)(36,43)(39,44)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=g,d^2=g,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, 2668122);