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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1859 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)(6,9)(10,12)(13,14,18,15)(16,21,24,19)(17,20,26,22)(23,28,25,27)(29,30,34,32)(31,35,39,37)(33,40,41,36)(38,43,44,42);(7,11)(10,12)(14,15)(19,21)(20,22)(27,28)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,19,24,21)(17,22,26,20)(23,27,25,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,42)(40,43)(41,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,43,44,42);(5,7)(6,10)(8,11)(9,12)(13,17,18,26)(14,20,15,22)(16,23,24,25)(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=h,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,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, 5524345);