Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 1048576 |
| 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)(5,6,10,12)(7,8,9,11)(15,21)(16,23)(19,24)(20,25)(29,30)(31,37)(32,38)(33,41)(34,35)(36,43)(39,40)(42,44);(5,7,10,9)(6,8,12,11)(14,18)(19,24)(20,25)(26,28)(30,35)(31,38)(36,41)(39,44);(5,8,10,11)(6,7,12,9)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31)(30,32)(33,39)(34,38)(35,37)(36,40)(41,42)(43,44);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,31)(33,42)(34,37)(35,38)(36,44)(39,41)(40,43);(5,9,10,7)(6,11,12,8)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,43)(30,36,35,41)(31,39,38,44)(32,40,37,42)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fg,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4326608);