Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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)(5,6,9,10)(7,8,11,12)(13,14,18,17)(15,19,23,22)(16,24,26,20)(21,28,25,27)(29,30)(31,33)(32,40)(34,35)(36,39)(37,41)(38,44)(42,43);(5,7,9,11)(6,8,10,12)(14,17)(19,22)(20,24)(27,28)(30,35)(32,41)(33,36)(38,42);(5,8)(6,7)(9,12)(10,11)(13,15,18,23)(14,19,17,22)(16,25,26,21)(20,28,24,27)(29,31)(30,36)(32,42)(33,35)(34,39)(37,43)(38,41)(40,44);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,20)(15,21)(17,24)(18,26)(19,27)(22,28)(23,25)(29,32)(30,37)(31,38)(33,43)(34,41)(35,40)(36,44)(39,42);(5,6)(7,8)(9,10)(11,12)(13,17,18,14)(15,22,23,19)(16,24,26,20)(21,28,25,27)(29,33,34,36)(30,31,35,39)(32,43,41,44)(37,38,40,42)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=fgh,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4407785);