Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 | |
1706 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)(6,9)(7,11)(13,14)(15,21)(16,18)(17,20)(19,22)(23,28)(24,26)(25,27)(29,30,34,32)(31,35,39,37)(33,40,41,36)(38,43,44,42);(7,11)(10,12)(14,16)(15,22)(20,24)(23,25)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,21)(17,25)(18,22)(20,28)(23,26)(24,27)(29,31)(30,35)(32,37)(33,38)(34,39)(36,42)(40,43)(41,44);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,14)(15,21,22,19)(17,24,26,20)(23,28,25,27)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(13,17,18,26)(14,20,16,24)(15,23,22,25)(19,27,21,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=fg,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4508121);