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)(15,19)(16,18)(17,24)(20,25)(21,23)(22,28)(26,27)(30,34)(31,37)(36,42)(39,40);(7,11)(9,12)(14,16)(15,21)(20,24)(22,27)(29,30,33,34)(31,35,37,38)(32,36,41,42)(39,43,40,44);(5,7,8,11)(6,9,10,12)(13,15,18,21)(14,19,16,23)(17,22,25,27)(20,26,24,28)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,43)(41,42);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,14)(15,19,21,23)(17,20,25,24)(22,28,27,26)(29,31)(30,35)(32,40)(33,37)(34,38)(36,44)(39,41)(42,43);(13,17,18,25)(14,20,16,24)(15,22,21,27)(19,26,23,28)(29,32,33,41)(30,36,34,42)(31,39,37,40)(35,43,38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=gh,d^2=fg,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, 2669098);