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 | |
1895 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)(8,9)(10,12)(13,14)(15,18)(16,21)(17,20)(19,24)(22,25)(23,28)(26,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(5,6)(7,11)(8,9)(10,12)(14,15)(17,25)(19,21)(23,26)(29,30,33,36)(31,34,39,37)(32,35,41,44)(38,42,40,43);(6,9)(10,11)(13,15,18,14)(16,21,24,19)(17,22,25,20)(23,28,26,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,24)(14,19,15,21)(17,26,25,23)(20,28,22,27)(29,31)(30,34)(32,40)(33,39)(35,43)(36,37)(38,41)(42,44);(5,7)(6,10)(8,12)(9,11)(13,17)(14,20)(15,22)(16,23)(18,25)(19,27)(21,28)(24,26)(29,32,33,41)(30,35,36,44)(31,38,39,40)(34,42,37,43)Polycyclic presentation:
2^9|b^2=gh,c^2=fgh,d^2=fgh,e^2=gh,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, 5544623);