Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1738 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)(7,11)(10,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30,34,35)(31,32,36,40)(33,41,42,37)(38,44,43,39);(5,6,9,8)(7,10,11,12)(15,21)(19,22)(23,27)(26,28)(30,35)(31,36)(33,42)(39,44);(5,7,9,11)(6,10,8,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31)(30,32)(33,38)(34,36)(35,40)(37,39)(41,44)(42,43);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32)(30,36)(31,35)(33,39)(34,40)(37,43)(38,41)(42,44);(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,33,34,42)(30,37,35,41)(31,38,36,43)(32,39,40,44)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4221512);