Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1734 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)(6,7)(8,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(30,34)(31,37)(32,40)(42,44);(6,7)(10,11)(15,21)(19,22)(23,27)(26,28)(29,30,33,34)(31,38,37,35)(32,41,40,36)(39,42,43,44);(5,6,9,7)(8,10,12,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,31,33,37)(30,35,34,38)(32,39,40,43)(36,42,41,44);(5,8)(6,10)(7,11)(9,12)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fg,d^2=h,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, 4375458);