Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(6,10)(7,12)(14,18)(16,23)(19,24)(22,27)(29,30,33,34)(31,35,37,42)(32,39,41,36)(38,44,40,43);(5,6,9,10)(7,8,12,11)(14,18)(15,21)(20,25)(22,27)(30,34)(31,37)(36,39)(38,40);(5,7)(6,8)(9,12)(10,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,30)(31,35)(32,36)(33,34)(37,42)(38,43)(39,41)(40,44);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,31,33,37)(30,35,34,42)(32,40,41,38)(36,44,39,43);(5,8,9,11)(6,12,10,7)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,32,33,41)(30,36,34,39)(31,38,37,40)(35,43,42,44)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,c^2=fg,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4775842);