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 | |
1773 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)(7,11)(8,10)(9,12)(13,14)(15,20)(16,22)(17,21)(18,19)(23,25)(24,28)(26,27)(29,30,33,36)(31,34,38,42)(32,39,40,35)(37,43,44,41);(5,7)(6,11)(8,9)(10,12)(14,19)(15,22)(21,27)(23,28)(29,31,33,38)(30,34,36,42)(32,37,40,44)(35,41,39,43);(6,8)(7,12)(13,15,18,22)(14,20,19,16)(17,23,26,28)(21,25,27,24)(29,31)(30,34)(32,37)(33,38)(35,41)(36,42)(39,43)(40,44);(1,2)(3,4)(5,8,10,6)(7,9,12,11)(13,16,18,20)(14,15,19,22)(17,25,26,24)(21,28,27,23)(29,30)(31,34)(32,35)(33,36)(37,41)(38,42)(39,40)(43,44);(5,9)(6,7)(8,12)(10,11)(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,25)(22,28)(29,32,33,40)(30,35,36,39)(31,37,38,44)(34,41,42,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=f,d^2=fg,e^2=fgh,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, 5533282);