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 | |
1758 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)(5,6,8,10)(7,11,12,9)(13,14,18,17)(15,19,23,22)(16,24,26,20)(21,28,25,27)(30,34)(32,40)(35,37)(38,44);(14,17)(19,22)(20,24)(27,28)(29,30,33,34)(31,35,39,37)(32,41,40,36)(38,43,44,42);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,23)(14,19,17,22)(16,25,26,21)(20,28,24,27)(29,30)(31,37)(32,36)(33,34)(35,39)(38,43)(40,41)(42,44);(5,7,8,12)(6,9,10,11)(13,16,18,26)(14,20,17,24)(15,21,23,25)(19,27,22,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,44)(41,43);(13,17,18,14)(15,22,23,19)(16,24,26,20)(21,28,25,27)(29,32,33,40)(30,36,34,41)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fh,d^2=i,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, 4252306);