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 | |
1654 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)(5,6,10,8)(7,11,9,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,30)(31,35)(32,34)(33,40)(36,39)(37,41)(38,43)(42,44);(5,7,10,9)(6,11,8,12)(14,18)(16,25)(19,21)(22,26)(29,31,34,39)(30,35,32,36)(33,38,41,44)(37,42,40,43);(5,8,10,6)(7,11,9,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,25)(26,27)(29,31)(30,36)(32,35)(33,38)(34,39)(37,43)(40,42)(41,44);(1,2,3,4)(5,9,10,7)(6,11,8,12)(13,15,17,23)(14,19,18,21)(16,26,25,22)(20,28,24,27)(29,32,34,30)(31,35,39,36)(33,40,41,37)(38,42,44,43);(13,16,17,25)(14,20,18,24)(15,22,23,26)(19,27,21,28)(29,33,34,41)(30,37,32,40)(31,38,39,44)(35,42,36,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=fgh,d^2=i,e^2=fg,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, 3044360);