Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1762 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)(7,9,11,12)(13,14,17,18)(15,19,21,22)(16,24,25,20)(23,28,27,26)(30,31)(32,39)(36,38)(40,41);(7,11)(9,12)(14,18)(15,21)(20,24)(23,27)(29,30,34,31)(32,35,39,37)(33,36,42,38)(40,43,41,44);(5,7)(6,9)(8,11)(10,12)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,28)(25,27)(29,31,34,30)(32,35,39,37)(33,38,42,36)(40,43,41,44);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,22)(16,24)(17,18)(19,21)(20,25)(23,26)(27,28)(29,32,34,39)(30,35,31,37)(33,41,42,40)(36,44,38,43);(13,16,17,25)(14,20,18,24)(15,23,21,27)(19,26,22,28)(29,33,34,42)(30,36,31,38)(32,40,39,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2668121);