Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1616 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)(5,6,9,8)(7,10,11,12)(13,14,18,19)(15,20,24,27)(16,17,26,21)(22,28,25,23)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(7,11)(10,12)(14,19)(17,21)(20,27)(23,28)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(5,7,9,11)(6,10,8,12)(13,15)(14,20)(16,25)(17,23)(18,24)(19,27)(21,28)(22,26)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(1,2,3,4)(5,6,9,8)(7,10,11,12)(13,16,18,26)(14,21,19,17)(15,22,24,25)(20,23,27,28)(29,30)(31,37)(32,40)(33,36)(34,39)(35,41)(38,42)(43,44);(5,8,9,6)(7,12,11,10)(13,17,18,21)(14,16,19,26)(15,23,24,28)(20,25,27,22)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=h,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 3050522);