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 | |
1730 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)(5,6,9,8)(7,11,12,10)(13,14)(15,20)(16,24)(17,22)(18,19)(21,26)(23,28)(25,27)(31,37)(32,39)(34,40)(35,42);(14,19)(17,27)(20,28)(21,24)(29,30,33,36)(31,34,37,40)(32,35,39,42)(38,41,43,44);(1,2,3,4)(13,15)(14,20)(16,25)(17,21)(18,23)(19,28)(22,26)(24,27)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(5,7,9,12)(6,10,8,11)(13,16,18,26)(14,21,19,24)(15,22,23,25)(17,20,27,28)(29,31,33,37)(30,34,36,40)(32,38,39,43)(35,41,42,44);(5,8,9,6)(7,11,12,10)(13,17,18,27)(14,22,19,25)(15,21,23,24)(16,20,26,28)(29,32)(30,35)(31,38)(33,39)(34,41)(36,42)(37,43)(40,44)Polycyclic presentation:
2^9|a^2=fhi,b^2=h,c^2=g,d^2=i,e^2=hi,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, 4286527);