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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1702 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)(5,6,9,10)(7,8,11,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30)(31,32)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43);(5,7,9,11)(6,8,10,12)(15,21)(19,22)(23,27)(26,28)(30,35)(31,37)(36,43)(38,44);(6,10)(7,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31,34,37)(30,32,35,40)(33,38,42,44)(36,39,43,41);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32)(30,31)(33,41)(34,40)(35,37)(36,44)(38,43)(39,42);(5,8)(6,7)(9,12)(10,11)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,33)(30,36)(31,38)(32,39)(34,42)(35,43)(37,44)(40,41)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=f,d^2=fgh,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, 5539392);