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 | |
1622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,27)(26,28)(29,30,33,36)(31,37,38,34)(32,35,40,42)(39,43,44,41);(2,4)(5,7,9,6)(8,11,12,10)(14,15)(16,23)(17,25)(27,28)(31,38)(32,40)(34,37)(35,42);(1,2)(3,4)(13,15,18,14)(16,19,23,21)(17,22,25,20)(24,27,26,28)(29,31,33,38)(30,34,36,37)(32,39,40,44)(35,41,42,43);(5,8,9,12)(6,10,7,11)(13,16,18,23)(14,19,15,21)(17,26,25,24)(20,28,22,27)(29,32)(30,35)(31,39)(33,40)(34,41)(36,42)(37,43)(38,44);(13,17,18,25)(14,20,15,22)(16,24,23,26)(19,27,21,28)(29,30)(31,34)(32,35)(33,36)(37,38)(39,41)(40,42)(43,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=g,d^2=fi,e^2=f,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, 3067350);