Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1918 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)(7,10)(8,12)(9,11)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,30,33,34)(31,35,39,37)(32,36,41,42)(38,43,40,44);(5,7)(6,10)(8,11)(9,12)(16,24)(19,26)(22,28)(25,27)(30,34)(35,37)(36,42)(43,44);(6,11)(7,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,31,33,39)(30,35,34,37)(32,40,41,38)(36,44,42,43);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,30)(31,37)(32,36)(33,34)(35,39)(38,44)(40,43)(41,42);(5,8)(6,7)(9,10)(11,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)(29,32,33,41)(30,36,34,42)(31,38,39,40)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fgh,c^2=f,d^2=fhi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4991657);