Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1822 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,9,11,12)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(29,30)(31,32)(33,37)(34,35)(36,38)(39,40)(41,43)(42,44);(5,7,8,11)(6,9,10,12)(15,22)(17,25)(19,21)(20,24)(30,35)(31,38)(33,41)(40,42);(5,7)(6,9)(8,11)(10,12)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27)(29,31)(30,36)(32,35)(33,39)(34,38)(37,42)(40,43)(41,44);(1,2)(3,4)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,32,34,36)(30,31,35,38)(33,42,41,40)(37,44,43,39);(5,6)(7,9)(8,10)(11,12)(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)(29,33,34,41)(30,37,35,43)(31,39,38,44)(32,40,36,42)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=fg,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5516968);